I am being threatened with a lawsuit... Can the company prove that it
was me using the pirated version?
That is a legal question which no one here can answer (and you'd be best getting professional legal help with). However, several aspects of your question do touch on interesting aspects of security-related license DRM issues, so let's talk about those a bit.
Can a MAC address lead a company to the specific user?
In general, no. However, a MAC address can be used as a compelling indicator as to which computer something happened on. That's the reason that most "licensing software" used to use MAC addresses to "identify" the computer they were installed upon. Because MAC addresses are unique-ish, and because altering them is not brain-dead easy (and used to be quite hard), it's often treated as an identifier of the system that something happened on.
They have my phone number and an old email address for me. The
information they sent me is showing a MAC address and the specific
number of times the pirated software was used and the dates it was
It is very likely that files generated with this software include these things as metadata - again, primitive DRM practices do love the MAC address. If the company found files you generated, which had grabbed your email address (did you register a support account?) and included the times those files were operated on, then they'd have what they needed to gin up a letter for you.
But the lesson here is, every file you generate with software may include all sorts of metadata you didn't expect to be in there. Caveat emptor.