Why M-theory has only M2 and M5 branes? In string theory, depending on the type one considers, you get all kind of D-branes. What is so special in M-theory that only allows 2 and 5 branes?
The type of branes a theory has in constrained by the p-form fields it contains. There is only one supersymmetric gravity theory in 11d, and it has a single $p$-form with $p=4$, call it $G_4$. This means that the theory can possess brane solutions that either couple electrically to $G_4$ (M2 branes) or magnetically to $G_4$ (M5 branes).
To understand the dimensionality of an electric coupling, The $p$-form field corresponds to a $p-1$-form potential. $p-2$ of these directions correspond to the extended brane directions (if there are any), and the last one corresponds to time. Compare this with the familiar case of electromagnetism where $A_t$ couples electrically to a point particle's worldline, and the field strength is $F_2 = dA$. In the case of M-theory, this means that $G_4$ couples electrically to a two-dimensional object, i.e. M2 branes.
To see where the magnetic M5 branes come about, simple take the Hodge dual to obtain a 7-form: $\star G_4 = G_7$, and then $G_7$ couples (by the same arithmetic as above) to a 5-brane, i.e. M5 branes.
Oh, and just a comment, in the 10D string theories there are indeed many different branes, but not all kinds. They are constrained by the same counting. So for example, in terms of D-branes (not counting the NS branes), IIB only has odd-dimension branes and IIA has even ones.
In supergravity theory, 11d have 3 dimensional scalar object and 6 dimensional anti-symmetric tensor object. This allows that M-theory have M2 brane and M5 brane only.
(By studying on the following question, D branes Ns brane and p-branes, I found some interesting paper: Supermembranes, arXiv:hep-th/9611203. It seems to me that section "1.3 The brane scan" describes relevant information on this question. So I post it here. This review paper also covers other various dimensions such as 10d,9d, $\cdots$ up to 2d)