The vanilla 2.0 EXE has no support for separate character models. To fix the problem
you're having, open the "Md2.dat" file in Notepad and go to the line with the name of the character you want. Replace the first instance of the name with PLAY.
(For example,
SHADOW SHADOW.MD2 3.0 0.0
would become
PLAY SHADOW.MD2 3.0 0.0
)
I should mention that every single player will show up as that model regardless of them being a different character with this method.
If you don't want to do any of that, you can either use
SRB2 with some OpenGL fixes or
SRB2CE instead.
Those two are options if you don't want to mess with the Md2.dat and aren't some purist.