The sum of two indep binomials with the same parameter p is a binomial.
(With N=n_1+n_2 and same p.)
If the paramters are different I do not believe it's a binomial.
MGFs are unique and since we cannot recognize this one as a binomial, it isn't.
HOWEVER, you can get the mean and the second moment by differentiating with respect to t and setting t=0.
And from that you can get the variance, but I don't know why you would use MGFs to find the mean and variance in this problem.
This is a MOMENT generating function, so you can clearly get the MOMENTS by differentiating and setting t=0.
That's how it got its name.