Why are acids and bases are defined as both donors and acceptors of species in the acid-base theories?
These are primarily old theories that were kind of evidence or observation based. These are not completely correct theories as there are may loop holes in them. For eg. many types of compounds with significant acidic and basic characters cannot be predicted by these theories. Also, to a certain extent the Bronsted and Lewis concepts are complementary.Wikipedia says - ""


  A Lewis base is often a Brønsted–Lowry base as it can donate a pair of electrons to H+; the proton is a Lewis acid as it can accept a pair of electrons. The conjugate base of a Brønsted–Lowry acid is also a Lewis base as loss of H+ from the acid leaves those electrons which were used for the A—H bond as a lone pair on the conjugate base.


and


  In 1923, Lewis wrote An acid substance is one which can employ an electron lone pair from another molecule in completing the stable group of one of its own atoms. The Brønsted–Lowry acid–base theory was published in the same year. The two theories are distinct but complementary.


Basically, proton is a $H^+$ ion and we all know from our simple understanding that compounds that give $H^+$ or proton are acids and those that accept $H^+$ or proton are bases.