Let's look at the big picture here.
nmap <plug>test a123 defines a Normal mode mapping that enters Insert mode, appends "123" and then remains in Insert mode. (I'm assuming that that's a requirement or I'd suggest something a little different.) So a result of "123<Plug>test" makes sense.
Rather than look for some "trickery" to get around the Vim's rules I'd suggest that defining the
nmap M command to do exactly what you want to do is the best course of action. By which I mean...
nmap M <plug>test<esc><plug>test
Here's the problem, though. That prints "123123" but it sounds like you actually want to print "123a123". That's, um, kind of weird since you want that "a" to be treated as Normal mode
append in the first case and the literal character "a" in the second case. For lack of a better word, that makes things a bit "hacky" and really prevents finding a solution that isn't itself a bit hacky.
Anyways, if that's what you want and not just an artifact of the example code you chose then a slight modification to the above mapping would be necessary:
nmap M <plug>testa<esc><plug>test