I hope this one (pun intended) post won't get ripped by the community. I wondered what are the most abstract ways to define unity element? Why is there a need for unity element in general? Is it just to define reciprocal elements in algebraic structures?
What types of $1$'s do we have in math and how they are connected?
What, exactly, do you mean by "unity"? There can be several different ways that word can be used be used. I suspect you are referring to the "multiplicative identity". defined as that member. e, of an algebraic structure, in which "multiplication" is defined, such that for any "x", e*x= x.
In order to need to distinguish between "additive" and "multiplicative" identities, so that we need to say "unity", we need a "ring" or "field". Indeed, the existence of a unity is, simply, part of the definition of "ring" or "field" You can have algebraic structures that do not have a "unity" but they simply aren't very interesting! Too many important algebraic properties depend on the existence of "unity".