No, apparently there isn’t. Stick does have its origin in the Germanic language family, however from what is nowadays in German “Stecken” for it’s penetrative aspect. (Yeah no kidding here, that’s what the etymology dictionary said)

Edit: just read the entry to “Stück” apparently there’s the idea of “Stückelung” as in parts of a larger whole, which coincides to the idea of a “Stock” (stick from a tree) being a separate part of the larger entity “tree”. Going by that logic I can see a similarity