They sound alike. Are they cognate historically? Morphologically, is よ in both cases a particle or part of the morpheme in -ましょう?
I think that this verb is the only one I've seen in Japanese that has so many definitions. とる and つく have multiple definitions as well (quite a bit IIRC). But not as much as 掛ける. In any case, I am ...
I noticed that verb ending syllables cover all of -u syllables (る,く,ぐ,す,つ etc) except ず,づ, ふ, ぷ, しゅう, ちゅう and じゅう. I suspect that ず is reserved for the negative conjugation thus no plain form verb ...