This is pretty much standard behaviour for all PS5 versions, they are separate SKUs with their own trophy lists. It is also desired behaviour for a lot of PS5 owners (although some may prefer previously earned ones to auto pop again) as many value being able to play again for a new set of trophies.
So i did the update for the ps5 edition of Skyrim special edition. Started game and was asked to spend $20 for anniversary updates. Paid the Bethesda Tax. Downloaded the expensive mods. Played for 30 min and found a goblin and a backpack that increases weight limit. So far i like it. Oh, the load times are like 1 sec now on ps5 version. Very nice.
Finally got a game breaking bug. Not a single one of my saves will load. The game boots, the loading is normal, black screen... Music playing. But that is it. Tried resetting my ps5, tried different saves, all the same thing.
And no, 99% of the time it’s just the base game for new releases and DLC is a separate purchase. E.g all the Forza’s, Doom games, Outer Worlds etc. The only new game from MS with DLC that hit Game Pass was Hive Busters.
The Legendary Edition is on Gamepass. It has all the official DLC. The Anniversary Edition with all the extra MODs must be paid for separately.
The Gamepass version provides everything you need for the complete Skyrim experience. The Anniversary Edition just adds many fanmade mods. Many of them are good.
I don't know who gave you the impression that Gamepass games include all DLC, but that is as Banjo64
put it, it's almost never the case that the Gamepass version will include DLC. In fact, I have seen many instances where the DLC costs so much that it's almost worth it to purchase the entire game separately. It's one of the ways that MS has the option to make good money off of Gamepass. For example, the Fallout 4 season pass is still $49.99 on Gamepass/Xbox.