Is a RSA-signature of some identifying data a safe way to implement a license key?
That is not a bad method of doing a license key; an attacker would certainly be unable to generate a signature that would validate with his computer's window product key.

On the other hand, this approach may be overkill. The easiest way to attack this system would be to modify the program to skip (or ignore) the signature validation; hackers have both the experience and the tooling to do this fairly easily.  I don't believe that there are any easily implemented license key approaches that make that attack more difficult; however, there are certainly other approaches that would be easier to implement.

Since you asked for cryptographical advice, here is some:


If you do use this approach, you might as well make sure you get the details correct; you use a known-good padding scheme (say one of the ones listed in PKCS #1).
You ask whether the window product key is long enough.  The answer is "Yes, it is".  Actually, the length of what you sign is not important to the security of the signature method; it just needs to be long enough to be unique.