My question is: Is there a max number of types of elementary particles predicted in advanced physics theories such as string theory? What are the reasons for this?Are the arguments purely mathematical?

Definitely, at least in String theory, there is no such finite number. In string theory, the closed string mass spectrum is given (in unnatural units, although it would work in natural units, too) by:

$$m=\frac{2\pi T\ell_s}{c_0^2}\sqrt{N+\tilde N-a-\tilde a}$$

Where $a,\tilde a$ are the left- and right- moving normal ordering constant. Here, the number operators $N,\tilde N$ can be ANY integer or half-integer. So, they vary from 0 to $\infty$ and thus, so does the mass $m$. In other words, the mass spectrum is infintely large.

And since each mass correponds to a different particle, there is an infinite "particle spectrum"...