I bought the game. I don't have to buy the game again. I stick in my PS4 game disc into the PS5 and it plays. If you give me a PS4 game for free and I stick it into the PS5 it still will work. You paid for it. My Xbox or Xbox 360 or Xbox One games that I have paid for or got free from a friend that he paid for will work when I put them in without having to pay again. That's backwards compatible. As long as I'm not paying again to play these same games it's backwards compatible.
That's not the definition of backwards compatible.
According to the definition online XBox does not have BC
(of computer hardware or software) able to be used with an older piece of hardware or software without special adaptation or modification.
But if you want to call what XBox does BC, then ps2 games with adaptation/modification free or not are BC on ps4 / ps5 as well.
You can play old PC games you have bought before on new systems by doing some work yourself (Configuring DOS box for example) or you can buy the package for a small fee from GOG (rebuying the game) who do that compatibility work for you.
Sony charges more to fit the old ps2 games into the emulator, but in essence it's the same and still the same base software.
Of course then the line blurs with remasters, more and more modification until you get into remakes which makes it a 'new' game.