Does mach keep track of tools capable of plunging (center cutting or not center cutting) for milling?

No, you can have Mach keep a note of which tool number is loaded but Mach doesnt know or care what type of tool it is.
Hood

Would that not cause problems if you don't have a center-cutting mill in place and forget the program calls for drilling?

Yes it would but how can Mach be responsible for that. Even if you defined a tool and Mach would not do a move if the tool defined couldnt do that move but how would Mach know you actually have put that tool in there?
The operator has to take some responsibility
Often CAM programmes put a description of the tool in the code beside the tool call, that way the operator can check through to see what the code requires.
Hood