If you don't need the stdout messages, you could do:
make 2>&1 >/dev/null | tee your_file
Note that the order (ie first 2>&1) is important: 2>&1 must be put BEFORE >/dev/null, otherwise your error messages will be sent to /dev/null.
Alternatively, you might consider named FIFO pipes. See "man mkfifo".
And there is no "stderr counterpart" to tee. tee just copies whatever it gets on it's stdin to it's stdout and to a file. What it gets on it's stdin can come from anywhere (another programs stdout, stderr, a file, ...).