The most successful example for a blockchain game is "Axie Infinity" which is something of a Pokémon-like. It used blockchain technology to track the unique "Axies" (= Pokémons) which could be traded and sold. So the blockchain was used to do just that.
The big promise of games like "Axie Infinity" was "play-to-earn", games which allowed players to earn money through playing the game, such as trading in "Axie Infinity".

Could this only be done with blockchain? No. All of that could technically also be achieved by other means.
So ultimately all of the talk about the blockchain was mostly PR and a way to distinguish the game. Nobody would have cared about it, if it had not had this feature. Which is very representative of all blockchain games.

The talk of how blockchain technology would allow players to transfer items from one game to another; or create unique characters which could be transfered between games; etc. have always been pipe dreams, They would have required a level of cooperation between publishers and developers of games that was simply impossible to achieve.

As a footnote: The use of blockchain in "Axie Infinity" ultimately resulted in an in-game economy that was largely a pyramid scheme. The game is still there, but the economy imploded and most players only ever made cents, if they earned anything through the game at all.