Standard non-metallic 8 gauge wire is only allowed to handle 40 amps. If you want to put 50 amps to the outlet you need more expensive wire (most likely 6 gauge, but there are other forms of 8 gauge that are approved for 50 amps). Since most stoves only need 40 amps electricians will run non-metallic 8 gauge by default and put a 40 amp breaker on it - then since there isn't such a thing as a 40 amp outlet they have a 50 amp outlet on the other end.

Yeah the main relationship that matters is the breaker to wire.

There is a fun semi related thing. Can you put 15 amp outlets on a 20 amp circuit? The answer is yes you can, but only if there is at least two receptacles, and a standard duplex outlet counts at two receptacles.

Which is another reason you don't want to oversize a breaker; you can have ten, twenty two-receptacle outlets on a single circuit, and if lots of stuff was plugged in you could easily draw tens or hundreds of amps, which would melt normal Romex pretty quickly.

This is an exception to the electric code, not the rule.

Almost every outlet must be paired with the appropriate wire size and breaker. A 30A outlet must be backed with a 30A breaker (no exceptions!) and wire must be at least capable of carrying 30 amps. There's also restrictions on wire size based on the outlet/breaker, since certain breakers/outlets cannot handle certain wire sizes. For instance, NEMA 5-15 outlets generally can't support wire sizes above 12 gauge, so you're limited to 12 or 14 gauge only.