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?
The short answer to your initial question is no. The historical/etymological spelling of 〜ましょう was 〜ませう, which is the expected form of the volitional, since the irrealis (未然形) stem of 〜ます is 〜ませ. Regular sound change explains the rest: せう becomes しょう.
The same story applies to the consonant-stem (五段活用) verbs: the old spelling for 行こう was 行かう (as seen here) etc. However, something strange happened to the vowel-stem (一段活用) verbs; instead of 見む → 見う → **みゅう or せむ → せう → **しょう, these were replaced by the forms with 〜よう we know today.