For example: assume that at some point, Super Mario Maker (which is effectively solely an online service) is going to get a release for the Nintendo Switch. Like with the 3DS version, a hypothetical Switch version would probably get its own global level-uploads database, partitioned off from that of the Wii U version.
Let's say Nintendo then, eventually, shut down the servers for the Wii U version of the game, while keeping the servers for the Switch version online.
Under the "separate games" interpretation, the backend for the Wii U version of Mario Maker would probably become abandonware as soon as it was shut down. People could set up their own compatible backend to put their level data back up and keep playing SMM on their Wii U's, and it'd be entirely legal.
But under the other two interpretations, it wouldn't be legal.
And keep in mind that, even if the client was a complete re-write, the servers would likely consist of 99% the same code between the two "generations" of the game. You could say that the Wii U version's servers "became" the Switch version's servers. One "evolved into" the other, without the Wii U version's ever actually "shutting down." They just, at some point, dropped compatibility for speaking to the old Wii U client software.
Given that what you bought in this case is the client software... would the previous paragraph matter? It's entirely an implementation detail hidden to you.