Basically any 5th wheel RV, and some large TTs are going to have a NEMA 14-50 plug and require a 50A 4 wire outlet. They do not have any 240V usage, they simply have 2 sides of the distribution panel, one for each 120V leg.
Most TTs are going to have a 30A, 120V plug.
Virtually everything in the OP is wrong. 10AWG is not rated for a 50A connection, and should be upgraded to 6AWG.
Second, you need a 3 wire plus ground, so NM-b 6/3 for the correct plug.
If the 10/2 is wired to a 4 wire 50A outlet with a 50A breaker, that is a major hazard and needs to be fixed ASAP. The likelyhood of an RV drawing 50A on either leg is pretty low, but could still potentially happen, and cause a fire.
The other hazard is with the current installation, the ground is now acting as a neutral, which again is dangerous.
In addition to the lack of a neutral and wrong wire size, I believe the breaker needs to be a GFCI.
I believe that is only per the 2020 code, so in most states it wouldn't be required yet.