Well Thiago had innate natural ability. Sometimes (and still does at 30 - lol) to the point where the basics would suffice better. But more often than not he gets away with it and become a world class midfielder.
Anyways, to me, (and I have only seen Gavi 5 or so times in full so take this with a pinch of salt) Gavi too seems that this is just naturally how he players and not something he groomed copying previous players.
But yeah, I recall watching Thiago in his formative years and he was a very special player. Gavi seems closer to a young Alena (not in terms of style but general youth level and hype) who also was having friendly debuts around 16/17 years old.
Thiago is an artist, and I didn't see Gavi as one from the 2 or 3 games I saw, and that's what made me a bit puzzled by the comparison.
But after your explanation, I think I can agree, he seems to be a bit of a natural talent, which Thiago and obviously Messi for example, also was, but in a bit different ways. Iniesta as well. He has a good sense of how to position himself and how to keep the ball, some of it seems intuitive, natural.
I think however, from that youth tournament game, that he looks to have a bit of "Cristiano" in him as well, in the sense that some of his dribbles seemed pre-planned rather than thought up in the moment. He waits for moments to do pre-planned stuff, and in my opinion he waits for too long...
Both in the youth tournament, where he was the big star, and in the preseason so far, he has been a bit too conservative in my opinion - and I have a big preference for players who don't rush things. I think he is too quick to pass it away rather than hold on to the ball and look for an opening. But let's find out more about him!