Try with fresh Kernel ?

$Version
(*"13.2.0 for Microsoft Windows (64-bit) (November 18, 2022)"*)

Remove["`*"];ClearAll["`*"];
Solve[m g Sin[\[Theta]] == m v^2/r && 0 <= \[Theta] <= \[Pi], \[Theta], Reals]

(*{{\[Theta] -> ConditionalExpression[\[Pi] - ArcSin[v^2/(g r)], Or[
And[
Inequality[-(g r)^Rational[1, 2], LessEqual, v, 
       LessEqual, (g r)^Rational[1, 2]], g > 0, r > 0], 
And[
Inequality[-(g r)^Rational[1, 2], LessEqual, v, 
       LessEqual, (g r)^Rational[1, 2]], g < 0, 
      r < 0]]]}, {\[Theta] -> 
   ConditionalExpression[ArcSin[v^2/(g r)], Or[
And[
Inequality[-(g r)^Rational[1, 2], LessEqual, v, 
       LessEqual, (g r)^Rational[1, 2]], g > 0, r > 0], 
And[
Inequality[-(g r)^Rational[1, 2], LessEqual, v, 
       LessEqual, (g r)^Rational[1, 2]], g < 0, r < 0]]]}}*)

Try:

Solve[m g Sin[\[Theta]] == m v^2/r && 0 <= \[Theta] <= \[Pi], \[Theta], Reals]

(*Simple solution on MMA 13.2.0*)