On the one hand, in street-talk, it is possible to omit the possessive suffix if the possessive adjective is present. So you can easily hear things like "benim eve geçelim" (instead of "benim evime") or "senin araba kaç model?" (instead of "senin araban").
On the other hand, there are certain expressions that are only used in this form. "Bizim el" or "bizim eller" in your song means "bizim köyümüz/memleketimiz". El in this literary usage refers to one's hometown. (Actually, more like "benim köyüm", as it is common to use biz to refer to oneself and one's extended family.)
In this context, bizim elimize would literally mean "to our hands", and would lose the meaning of hometown.
