The function field analog of the theorem of Mazur-Kamienny-Merel (giving a universal bound in $[K:\mathbb{Q}]$ for the size of the $K$-rational torsion of an elliptic curve) is immediate just from the fact that the modular varieties involved happen to be algebraic curves with gonality going to infinity with the level. In fact the bound here turns out to be polynomial in the gonality of the function field, something which is conjectured to hold but not known in the original setting.

Regarding the higher dimensional variant of the problem over function fields, I am only aware of a 1989 paper of Nadel proving that a non-constant p.p. $g$-dimensional abelian variety over a one-dimensional complex function field cannot have a full rational $n$-level structure for $n \gg_g 0$. This however was a long time ago; any progress since then? The simplest statement here is the uniform boundedness of the group of $k(T)$-rational torsion of a traceless abelian surface over $k(T)$.