There isn't really any connection between the Hungarian "ssz" sound and the spelling ſ or ß.
The question about the origin of ssz in Hungarian is interesting. What I have gathered so far is that ssz in Hungarian is fairly infrequent, and may be disproportionately more frequent in loanwords (where it seems to mostly be derived ultimately from "ss" in Latin or in Romance languages), but it was not introduced to the language by loaning. However, I also am not sure if there are any examples of it being inherited intact from some Uralic geminate consonant: I have the impression that in native words, it would originate from simplification of some other sequences. I need to do more research about this.
Also, as far as I can tell, there doesn't seem to be anything particularly special about ssz as compared to other long consonant sounds like tt, kk, or ccs.
To refer to spellings, I'll use the angle-brackets convention, so ⟨s⟩ is a spelling, [s] is a sound, and /s/ is an element of a sound system.
The spelling ⟨ſ⟩ is not "long" in the sense of phonetic length
The letter-form ⟨ſ⟩, sometimes called "long s", was as far as I know never used to represent a phonetically lengthened s sound. The term "long" refers to its shape, not its pronunciation: it is vertically elongated. It was used historically as a non-final variant lowercase form of the letter S in the spelling systems of a number of languages. Because of rules about how to write certain kinds of words, like compounds, it occasionally represented a distinct pronunciation from "short/round s" (e.g. in German, a famous example is Wachſtube /ˈvaxˌʃtuːbə/ vs. Wachstube /ˈvaksˌtuːbə/), but you can see from this example that it was not a matter of a phonetic or phonemic length distinction.
German ⟨ß⟩ is supposed to have originated as a ligature, the first element of which was ⟨ſ⟩. Since German spelling historically used both ⟨ß⟩ and ⟨ſ⟩ as contrastive elements, it's confusing to call ⟨ß⟩ a "German long s". Wikipedia says that ß is called Eszett or scharfes S in German, and eszett or sharp s in English.
Hungarian has a two-way phonological contrast between voiced and voiceless consonants for most obstruents, so /s/ ⟨sz⟩ and /z/ ⟨z⟩ are distinct phonemes.
It also has a distinct phonological contrast between short/basic/single and long/double/geminate consonants, as with [s] ⟨sz⟩ vs. [sː] ⟨ssz⟩. The phonemic analysis of long/geminate consonants in Hungarian seems to be controversial, with some linguists analyzing them as mere sequences (Ham 2001, p. 101). In the rest of this post, I will use the term "geminate" to refer to things like ⟨ssz⟩. Hungarian spelling differentiates geminate ⟨ssz⟩ from doubled ⟨szsz⟩ (the latter is found in compounds) and I read somewhere (sorry, I forget where) that there may be slight differences in pronunciation also between ⟨ssz⟩ and ⟨szsz⟩, but that the realizations overlap.
It seems that just about every consonant in Hungarian can be geminate, although some consonants are more commonly found as geminates than others. (According to Ham 2001, p. 102, all Hungarian consonants are found geminate except for [h] and [ʒ]. It seems that in fact, geminate /h/ exists, but is realized as [xː]. Geminate /ʒ/ seems to exist in inflected forms, at least, such as rozzsal (the instrumental case of rozs), even if it doesn't exist in the lemma form of any word.)
No consonant is found very frequently as a geminate: the table of frequencies in §188.8.131.52 of Grimes 2009 indicates that in dictionary forms, every consonant phoneme is found more than 92% of the time as a single or basic rather than a geminate consonant. Although [zː] seems to be rare, this frequency table indicates that words with [zː] do exist, so we can say that there is a four-way contrast between [s], [sː], [z], [zː].
⟨ssz⟩ in native words
Unfortunately, I'm not familiar with the sound changes that applied between Proto-Uralic and Hungarian. Here are some words with ⟨ssz⟩ from Wiktionary's list of "Hungarian terms inherited from Proto-Finno-Ugric", and their etymologies as given by Wiktionary:
"Borrowed consonant lengthening" in Hungarian
I found a PDF by Bálint Huszthy with some relevant information about the occurence of geminate consonants in Hungarian. Not too surprisingly, geminates have restricted distribution compared to single consonants, and in native vocabulary geminates are rare and have a low contrastive load with singleton consonants. Because of this, geminates may have a role in Hungarian as a characteristic element of "foreign vocabulary". It seems to be debated to what extent gemination in foreign vocabulary is related to the influence of German dialects specifically. Huszthy says that Nádasdy (1989) suggested borrowing from German dialects with gemination led to analogical extension of gemination to other foreign words. But Huszthy (2016) says that "There must be other motivations of ["borrowed consonant lengthening"] beyond language contact" and suggests that it is related to constraints about prosodic structure.
This phenomenon in Hungarian reminds me a bit of how Japanese often has geminate consonants in loanwords, e.g. Bahha from German Bach or kyappu from English cap.
⟨ssz⟩ doesn't seem to be too special compared to other geminate consonants
Huszthy does not indicate any special status for ⟨ssz⟩ as opposed to other geminate consonants. The PDF includes a number of examples with geminate consonants other than ⟨ssz⟩: both ones where the doubling might be taken at least in part from the spelling used in the source language ("desszert, gleccser, hobbi, koffer, kollázs") and ones where the doubling is not present in the spelling of the source language ("dajjer, dopping, maffia, Guttenberg,
Apparently, voiced fricatives are rarely found geminate, whether in native vocabulary or in loanwords (Magyar 2016, p. 290).
Hungarian ⟨ssz⟩ in words from German seems to correspond to ⟨ss⟩, not ⟨ß⟩
I looked through Wiktionary's list of Hungarian terms derived from German. All of the words in this category with ⟨ssz⟩ in Hungarian seemed to be derived from German terms spelled with ⟨ss⟩, not ones spelled with ⟨ß⟩. A number of the examples seem to be Latinate (e.g. szecesszió, agresszió, asszisztens, asszociativitás) so influence from some kind of Latin seems somewhat plausible to me (either from Latin spelling, or from some tradition of Latin pronunciation that maintained distinctly long consonant sounds in this kind of context).
Standard German does not have phonemically "long" consonant sounds
The usual linguistic description of Standard German does not include any length contrasts for consonant sounds. Instead, Standard German is described as having a two-way contrast for obstruent consonant sounds where one consonant is "lenis" or "voiced" and the other consonant is "fortis" or "voiceless" (I think it may still be disputed whether voice is a relevant phonemic characteristic in the German consonant system, but I'm not sure).
I've read that "lenis"/"voiced" fricatives tend to be phonetically shorter in duration than "fortis/"voiceless" fricatives, but this difference in phonetic duration is not treated as a phonemic length contrast in standard German.
I'm most familiar with the two-way obstruent distinction being represented in phonemic transcriptions of Standard German by the use of IPA symbols for voiced vs. voiceless consonants: that is, /s/ is used to represent the fortis sibilant fricative, and /z/ is used to represent the lenis sibilant fricative, even though it is not supposed to be uncommon for German /z/ to be phonetically voiceless (at least partially). Standard German is described as having a process of "Auslautverhärtung" that applies to word-final (or syllable-final) obstruents and causes the contrast between lenis and fortis obstruents to be neutralized in perception in favor of fortis obstruents. That means word-final /z/ is supposed to be fortited, causing it to sound like /s/; likewise, word-final /d/ is supposed to be fortited, causing it to sound like /t/. I've read papers that suggest that this kind of neutralization may be incomplete in production.
Consonant sequences like /s.z/ may occur across syllable boundaries, but such a sequence is not considered to constitute a phonologically long or geminate consonant (compare how in English, /s.s/ can occur across syllable boundaries in compounds like 'housesitting", but English is not usually described as having geminate consonants in its inventory of consonant sounds).
Consonant length in historical and dialectal varieties of German
There are supposed to be dialects of German that do have phonemic length contrasts, but I don't know enough to summarize the situation. As mentioned above, it seems to be thought that dialects like this may have been the source of some Hungarian words with geminate consonants (not just ⟨ssz⟩ specifically, but also other geminates).
From a historical perspective, German ⟨ß⟩ mostly represents older /t/, via the "High German consonant shift". In German, the inherited /t/ sound is supposed to have developed in many contexts to a long fricative with an s-like quality; the linked Wikipedia article says that it has been suggested that it was specifically a laminal voiceless sibilant. The resulting sound in modern standard German is standardly transcribed /s/, and spelled ⟨ss⟩ after short vowels, and ⟨ß⟩ after long vowels or diphthongs (in older spelling systems, the grapheme ⟨ß⟩ was used more widely). I haven't actually found any Hungarian word with ⟨ssz⟩ derived from Germanic /t/, though.
The form ⟨ſ⟩ was used in English to represent the same sounds as ⟨S⟩ or ⟨s⟩: mostly /s/ and /z/, in some cases /ʃ/ or /ʒ/ in words with historical palatalization-related sound changes. It's a bit similar to the old use of ⟨v⟩ and ⟨u⟩ as contextual spelling variants (⟨v⟩ at the start of a word, ⟨u⟩ elsewhere). Spellings like ⟨congreſs⟩ don't tell you anything about the pronunciation: the word might have been pronounced differently in the past, but that's not related to how it was spelled.
English had consonant length a long time ago (consonant length distinctions are uncontroversially reconstructed as being present in Old English, although long consonants had a restricted distribution, and I have the impression that they were fairly infrequent and that the long/short contrast didn't have a very high contrastive load). Long/geminate consonants could be spelled with doubled letters (I don't know how consistent this was), as in cyssan [ˈkysːɑn], the Old English cognate of Modern English kiss. Contrastive consonant length in English is thought to have been lost at some point in the Middle English period, which is before Shakespeare's time.
Grimes, Stephen M. 2009. "Quantitative Investigations in Hungarian Phonotactics and Syllable Structure".
Huszthy, Bálint. "Unmotivated" consonant gemination in the Hungarian Foreign Accent
Magyar, Lilla. 2016. "Are Universal Markedness Hierarchies Learnable from the Lexicon? The Case of Gemination in
Ham, William. 2001. Phonetic and Phonological Aspects of Geminate Timing.