The statement that Greek "doesn't have" gemination is really just a shorter way of saying that gemination is not phonemic in Greek.
Phonemic status of a feature in a language is an indication of whether that feature "makes a difference" in that language: so to say that gemination is not phonemic doesn't necessarily mean that gemination doesn't occur, but just that whether a consonant is geminated has no bearing on the phonemes contained in the utterance, which in turn means, roughly, that the consonant being geminated instead of simple isn't perceived by speakers of that language as a different language sound, but merely as a different way of producing the same sound (it doesn't make a word different). The usual way to determine phonemic status is to check for minimal pairs, which means answering the question "Are there two words in this language that are the same, except for this one feature?". In Greek, there aren't any words that differ only in that one has a geminated consonant, and the other doesn't.
If a given feature doesn't have phonemic status in a language, it could well be that it never occurs in the language, but it could also be the case (and I think it often is, though I have no hard data) that it occurs allophonically. Allophones are sounds that are perceived by speakers of a given language as "the same" sound, just with pronunciation differences that have no bearing on the direct meaning. It doesn't necessarily mean one is free to choose any allophone among a set of sounds, either: often, there are discernible rules underlying the way an allophone is chosen, which can be rules that pertain to which sounds are nearby, or rules that have to do with higher-level features of an utterance, such as its prosody or the emphasis a speaker is trying to convey on certain parts of the utterance.
It's entirely possible that Greek, for example, tends to geminate consonants that come right before a stressed vowel: that would not surprising, as in many languages, vowels become longer in stressed syllables, even if vowel length isn't considered to have phonemic status. Gemination is simply lengthening of consonants, and in this case, consonant length and vowel length wouldn't be seen as features of their own, but as parts of what concurs to forming the "stress" feature.
It could also be that this only happens when the speaker is trying to impart a particularly strong sentence stress to the word: again, this is similar to how when one wishes to stress a word in English or other languages, they may pronounce the stressed vowel(s) in it longer and/or louder.
Alternatively, allophones can be purely in free variation: in the case of gemination, this means that a language that doesn't feature it phonetically will have consonants of different lengths in a random way, simply because the speakers won't be paying attention to the length of their consonants: it's not something that matters in the language.
These are just hypotheses I'm using as examples, as I don't really know when gemination occurs in Greek, although the hypotheses I gave roughly correspond to my suspicions, and in particular I feel that unvoiced stops, nasals and liquids lengthen when they occur at the onset of a stressed syllable.
What I do know is that I also hear gemination occurring when I hear Greek (my native language being Italian, which does have phonemic gemination, like Hungarian), but that it doesn't occur phonemically, which, to reiterate in simpler terms, means that whether it occurs or not is irrelevant to the literal meaning.
I've provided this answer without listening to the song, because I'm not a speaker of Greek and it probably wouldn't help to give my subjective impressions. I've listened to it now, and since I've heard Greek a few times before, and I can read it and understand it very superficially, for the little it's worth, I can say this doesn't sound like Greek with a Swedish accent to me, but just like regular Greek, with non-phonemic gemination in unsurprising places.
