First thing to note is that DC fast chargers output high voltage DC to the vehicles they are charging, bypassing the onboard charger. So although most are theoretically capable of charging many different cars at their voltage requirements, not all are, and that's a base requirement. In order to even charge the different Tesla models and their differing packs, the chargers are capable of varying voltage and current. So they are designed to be variable, but depending on how it's designed, most likely the vehicle needs to communicate the voltage and current charging requirements for it's state of charge.

Secondly, Tesla's chargers have unique connectors and unique communication with the vehicles. Not to mention, the architecture is designed in a way that no payment, identification, etc. happens at the supercharger. You just plug it in and everything else is handled on your vehicle touch screen and Tesla web account. So really, unless Tesla decided to license the tech and allow other vehicle brands to id and communicate with their chargers, it wouldn't be possible to charge at a supercharger.

Charging stations coordinate with the vehicle to determine the charge rate, so there's no restriction based on power. Other EVs technically could use the Supercharger network, though many of them wouldn't be able to charge at the SC's maximum rate. Tesla has decided to keep it exclusive for now.

There are actually other charging networks that have more powerful chargers than Tesla; for example, while the V3 SCs max out at 250 kW, Electrify America's most powerful chargers max out at 350 kW.