Tail Light or Brake/Turn Signal Light?
Reeyates, In your post title, you say your "brake light stays on after releasing the brake", and then in the body "the right rear tail light"... Does your rig have something like dual filament 1157 (or rendition of) bulbs or some newer thing like LEDs? When you speak of 'brake' and 'tail' lights, you're speaking of two separate parts of the bulb.
An 1157 has two filaments, a brighter, more intense one for the brake/turn signal (does both functions) chore and a lesser intense, not as bright filament for the "tail" light function.
The tail and marker lights typically come off ONE lead at your connector. Turn your tow vehicle's lights off and all the tail and marker lights of the trailer should go off.
The turn signal wiring from tow rig to trailer will have a separate wire for each of the right and the left side, and draw from the tow rig's right and left turn/brake wiring. Both brake and turn signal use the same filament in the bulb. Turn signal function overrides the brake light function, i.e., if you are braking and signaling for a turn, say right turn, only the left brake light will come on, the bright filament, while on the right side, the brighter filament is blinking indicating a right turn intended. Release the brake pedal with turn indicator still 'on' and the brighter filament on right side continues to blink, and the left bright filament should go out. If the 'tail' lights are on, along with tow vehicles driving, parking, or head lights on, they should continue to remain on until the tow vehicle's running lights are turned off.
If the right brake/turn filament stays on (Vs the 'tail' light) after releasing the brake pedal, I would suggest that the tow vehicle's right brake light should also be on at the same time, and 'hanging up' as your trailer's turn signal is. What could possibly cause this, if this is what is happening, I couldn't say as it certainly isn't 'normal'.
Now if it is indeed the 'tail' light (not so bright a filament) that is staying on, then all the other exterior marker lights as well as the left tail light should also be on as they all draw from the same incoming wire at the connector. Their functioning is independent of the brake/turn lights. It's possible you may have a cross feed from something else to the right brake filament but that's highly unlikely, nor does it sound like a problem with the brake light switch as when activated by depressing on the tow rig's brake pedal would send power to all the brake light filaments, tow rig as well as trailer's. Long diatribe of what should be a simple problem, though it sounds strange as described---hope you get it resolved.