Let's us try to find an identity element from this
form a*b = (a + b) / (1 + a b / c^2).

       a*e = (a + e) / (1 + a e / c^2) = a,
         where
       a is the velocity of observer B, measured by observer A,
         and
       e is the velocity of a moving body as measured by
       observer B.

       a + e = a + a^2 e / c^2,
       e = a^2 e / c^2,
       e vanishes. We can't find an identity element.
       Only that a^2 = c^2 for all a, which is a contradiction.

According to the meaning of relativistic composition of
velocities provided by you, the existence of an identity
element would mean that observer A would measure the
velocity of any moving body as being the velocity of observer
B. But, a moving body  and observer B are independent.
How is it that observer A always measure the same velocity a
for a moving body, regarless the velocity of observer B measured
by A?. SR can't deal with a simple identitiy velocity. There is
no identity velocity in SR, fella.