Yakuza 5 is SO good, cannot recommend it enough- as someone who did the Y0-Y8 gauntlet recently, it is long, but it's super worth it, and some of the earlier games are pretty short if you just do the main story. Since you were wondering, the release order is Y1, Y2, Y3, Y4, Y5, Y0, Kiwami 1 (Y1 remake), Y6, Kiwami 2 (Y2 remake), Y7 (Yakuza: Like A Dragon in the west), Y8 (Like A Dragon: Infinite Wealth in the west). 3-5 have gotten HD remasters, but not full Kiwami remakes. ("Kiwami" means "extreme", by the way, as the games themselves love to tell you.) 1 & 2 were PS2, 3-5 were PS3, 0 & Kiwami 1 were PS3 and PS4, and Y6+ have been PS4 (plus PS5 from Y7 on). Y5 was 2012.
If you do play Y5, I think this'll be a fun video to come back to, especially the first part. Daigo is indeed a name, but I'm of the opinion that this track is much more for/about the other person present when it plays, rather than Daigo. You'll just have to play peak fiction to find out who that person is and what the context is. ;)