And I can no more access the xBox live store to buy games for my Playstation than I can use Steam to buy games for my Switch than I can use the PSN store to buy games for my PC than I can use the Android play store to buy games for my iPhone. Nor do I get all of the games I bought on the Switch when I start up the Playstation, iPhone, or PC.
Several of those services (all?) allow you to purchase through someone other than directly through them. For instance several of my steam games came from humble bundles. Additionally, most people only have one phone. Ownership of multiple gaming platforms is common.
Your reasoning is about as sound as saying that Harry Potter is Star Wars. Sure at a superficial high level the story has similarities, but as soon as you look at any of the details you realize that they're actually quite different.