As much as I theoretically agree, I can immediately think of two problems:

1. The storefronts would have to communicate

It's against their own interest to do this. Imagine you buy all your games on Steam because of the sales (although the creators of the game of course decide the prices, but still) and then play them on your Xbox. No profit at all for Microsoft, yet they're the ones providing all the additional services like the actual game hosting, friends system, etc. It's not much by any means, but it does add up. The money all goes to Valve. You could even buy the games via the Steam mobile app if you don't even own a PC. Also, even if they were theoretically fine with this, even coordinating it would be a pain. Since you could put a game on the Google Play Store, the App Store, hell maybe even F-Droid, Epic Games, GoG, Steam, the Xbox Store, and the Play Station store, and I am absolutely certain I forgot multiple other options, all of them would need to be able to communicate and decide on if you actually own the game. This would be a logistical and technical nightmare.

2. Companies would just sell mildly different versions and claim it's a new game

You know how for example Undertale has a slightly special Nintendo Switch version where there's... I can't even remember, but I think it's an additional boss. That's just something small and cute, but let's go with the GTA example. I have played about five hours of 5 and dropped it, so excuse me if this isn't the best theoretical example, but let's say the PS5 and Series X/S get the base game. Then the PS6 and new Xbox get maybe five additional cars and the game they're selling is GTA 6 Expanded. Afterwards on switch (although by that time Nintendo's new console would've released) you get blue and red weapon skins or whatever and it's GTA 6 Switched Up. And then finally on PC you get the GTA 6 Ultimate Edition with expanded settings, better graphics, and maybe five more cars on top of those from GTA 6 Expanded. These are all technically not the same game, so you would not be able to claim them. Sure, you could argue they're similar, but where is the exact line? That's quite impossible to figure out - is it a cheated rehash or a mediocre remaster? Who knows