The Swedish sound you refer to is a marked variant of /iː/ commonly heard in Stockholm and often associated with ‘posh’ speakers. It differs significantly from the unmarked realisations of /iː/, to varying degrees.
It is essentially produced as a regular [iː], except that the body of the tongue is somewhat lowered, and the blade (or tip) is somewhat raised towards the front part of the alveolar ridge, just behind the upper front teeth. This gives it a quality that is not unlike [ɨ], just more fronted and usually closer (with a tighter gap between the tongue and the roof of the mouth). It is sometimes even close enough that it can reasonably be described as a fricative; generally speaking, the closer and more fricative-like it is, the posher it sounds to Swedes.
The Stockholm /iː/ has its closest point of articulation just behind the upper front teeth and may be categorised as dental. When particularly close, it is acoustically not dissimilar to English /ð/ (but produced behind the front teeth, not interdentally, so more accurately it is not dissimilar to an Icelandic /ð/).
Phonotactically, the sound is grouped with the vowels in Swedish, even though it is arguably consonantal in many cases.
The Danish sound you describe is the normal, unmarked realisation of /ð/, which is normally a very open and vowel-like approximant.
It is a notoriously complex sound in articulatory terms, generally described as a retracted, velarised, laminal, alveolar approximant [ð̠˕ˠ]. Its production involves raising the back of the tongue towards the velum while lowering the tip. In a typical production of the sound, the tip of the tongue rests idly on top of the lower front teeth, while the main part of the dorsum is slightly convex (bulging upwards) in relation to the general arc from the raised back to the lowered tip. If you pronounce a typical [ɰ] (velar approximant) but move the tip of the tongue out to rest on top of your teeth, that’s pretty much it.
Danish /ð/ thus has two places of articulation that are more or less equally close (though both are in fact quite open): the front of the dorsum with the (post)alveolum, and the back of the dorsum with the velum. It is most commonly categorised as an alveolar consonant, but it can be grouped with velars as well. The new Danish pronunciation dictionary by Ruben Schachtenhaufen goes so far as to transcribe it as an unrounded back vowel [ɤ].
In very careful, theatrical or old-fashioned speech, the front place of articulation may be narrower, resulting in slight frication between the blade of the tongue and the alveolar ridge, or between the tip of the tongue and the upper front teeth. This is not normal in regular speech, though.
Phonotactically, /ð/ is grouped with consonants (though limited to postvocalic position), even though it is really more vowel-like in many cases.
In the archetypal productions of these two sounds (i.e., marked, posh Stockholm speech vs regular, unmarked Danish speech), they are quite distinct and would not easily be mistaken for one another. Somewhat perversely, as described above, Swedish /iː/ is generally more consonant-like than Danish /ð/.
Even so, it is possible to create an environment where the two sounds may acoustically approach one another, if:
- the Swedish sound is produced in quite a non-posh way (i.e., with a larger gap between tip of tongue and postdental ridge) so as not to produce any frication and make a more vowel-like sound not dissimilar to [ɨː]
- the Danish sound is produced in quite a theatrical or old-fashioned way, with a narrower gap between blade of tongue and alveolar ridge
The former of these occurs naturally in some dialects intermediate between ‘posh Stockholm’ and Central Swedish, so you’re not unlikely to hear it in real life. The latter no longer occurs naturally in any dialects and is now only heard as a deliberate affectation, or in old movies from the ’60s.