>If people would host "xyz liked my tweet" on their own server, then everybody would claim that Billie Eilish liked all their tweets.

This problem is what public key verification is literally made to solve.

If Billie Eilish likes your tweet, her client signs the like with her private key and sends the result to your server. Other clients can verify the like is real by looking at the result you send back with the "like" and verifying it against her public key.

Edit: The person I'm replying to has edited their post at least 3 separate times in the past couple minutes, adding multiple new lines to say different things, so if this disagreement to it ends up making no sense, you know why.