Why do different gases have the same molar volume?
You have the right thought process; if two real gases have different sized molecules, they won't have the same molar volume. The issue is that your statement about gases at STP refers to ideal gases which by assumption are just point particles that only interact via elastic collisions. To get a better sense of the difference that considering the finite size of molecules and their electrostatic interactions can make, you will want to read up on the various real gas models used, such as the Van der Waals, Redlich-Kwong, and Peng-Robinson models. 

As a small side note, MaxW is correct that the definition of STP you are using is out of date so the molar volume at STP with the new definition is slightly higher. Its still quite common to see people accidentally continuing to use the old definition, so be on the lookout for that.