What is the variable $(MAKE) in a makefile?

From the docs:

The value of this variable is the file name with which make was invoked

It is useful in cases where to make some target you'd have to call its makefile, but you are doing some sort of dry run with -t (--touch), -n (--just-print), or -q (--question) flags. That behaviour would propagate recursively if ($MAKE) is used.

It's a recursive invocation of make itself, forwarding the -t, -n and -q options. This makes sense: you want the nested make invocations to run with these options as well.