"Uniform" Set Cover Approximation?
No, one can't achieve a better approximation ratio for your problem.  This follows by a simple padding argument.

Let $S_1^*,\dots,S_m^* \subseteq S^*$ be an instance of ordinary set cover (where the sizes of the sets are not restricted to be the same).  Let $n = |S|$.  Define $S^* = S \cup \{1,2,\dots,n\}$ where it is assumed that $1,2,\dots,n$ represent $n$ new symbols not found in $S$.  Also define $S_i = S^*_i \cup \{1,2,\dots,n-|S^*_i|\}$ and $S_{m+1} = \{1,2,\dots,n\}$.  Then $S_1,\dots,S_m,S_{m+1} \subseteq S$ form an instance of your uniform set cover problem; by construction, the sets $S_1,\dots,S_m$ all have the same size.  Moreover, the minimum cardinality cover for the original problem $S^*$ differs in size from the minimum cardinality cover for the uniform problem $S$ by at most one, since given any solution $\{i_1,\dots,i_k\}$ for the original problem we obtain a valid solution $\{i_1,\dots,i_k,m\}$ for the uniform problem, and vice versa, any valid solution for the uniform problem is also a solution for the original problem.

It follows from this reduction that uniform set cover has the same approximation factor as ordinary set cover.  Thus, all the standard approximability results for set cover carry over to your problem as well.