If you want to understand the phonetic nature of /ð/, in general, you need to look at recordings of native speakers of languages that have it. The corpus should be contextually varied (e.g. utterance-initial, utterance-final, intervocalic, and so on), and it should not include any "sound performances" like "it is pronounced [ððððð]". It should also only contain data from actual native speakers of the language in question. In the Wiki recording, it isn't even claimed that this is "in a language". Some dialects of Finnish do have [ð] but I don't know if the performer of the wiki recording is such a speaker. I would be somewhat happy to give my rendition of a Danish [ð], but there is no question that it's not a native production.
Every phonetic symbol stands for a range of pronunciations. I would say that Danish [ð] is different enough from [ð] of Icelandic, Kven, Saami and English that it deserves a distinct symbol. But that would imply the possibility of contrastiveness, which has not been established. Those pronunciations are all within the realm of normal variation for how [ð] is pronounced, across languages and speakers. It would be kind of hard for us to diagnose why you have these different feelings about specific performances of [ð], if that's what you're asking. I suggest that you narrow the question down to something much more specific.
The revision of the question is much clearer. We have known forever that there will be some measurable difference between any two performances of exactly the same word by the same speaker, and questions of sameness / difference in phonetics imply some scale of granularity in measurement. The idea that guides crosslinguistic phonetic comparisons is that languages may differ in what specific values a sound has, so that [i] in German may have a lower F1 than it does in English, and that could be a fact of some linguistic interest, but simple anatomical variation can explain why two speakers of the same language or of different languages could have different phonetic values for a nominally same phoneme.
I don't judge the three samples referenced in the OP to be particularly different, but one can substitute Danish ð vs. English ð, or various other pairs of sounds that are audibly and systematically different between two languages. Then the question is whether the IPA should add symbols to systematically represent that difference. But the IPA does not purport to represent all possible systematic sound differences. This is most clearly seen in the illustrations of the IPA, where vowels are positioned in a standard vowel space. You will notice that the symbol a is used very frequently, but the position that it occupies across languages varies considerably. Sometimes one finds a contrast in low vowels such as æ,a or ɑ,a which constrains the positioning of a, but without a contrast, one usually find that the low unrounded vowel is written a, even if it crosses into back vowel space.
Note that the Handbook of the IPA (1999: p. 27) states that "the International Phonetic Association has aimed to provide 'a separate sign for each distinctive sound; that is, for each sound which, being used instead of another, in the same language, can change the meaning of a word'". If a language is uncovered which phonemically contrasts two kinds of ð, Danish-style (approximant) and English-style (fricative), then one could rightly conclude that there should be two distinct symbols for representing that difference. Until that day (plus time for voting...), it should not. However, also note that there are diacritics that can be added to specify particular kinds of pronunciation detail, so that the Danish ð could be written as ð̞.
If one encounters a language with a voiced dental / interdental affricate, then d͡ð would be appropriate as a transcription. I would not include that particular performance of ððððð in the discussion, since this isn't an example of language, it's an example of meta-language and probably not even representative of normal speech for that person. However, there are similar issues that arise in the transcription of language, such as the lip action of Tokyo Japanese "u", or the labial protrusion of Shona orthographic sv, zv, formerly transcribed as ȿ,ɀ. If one encountered a somewhat labialized ð, this could be rendered with the seagull diacritic.