Letter of support for promotion to full professor from an "unusual" student

The department asked you to write because they want your opinion. Of course they also asked others. If they do not ask you specific questions, then just give the reasons for

I am super happy to be able to support her/his promotion because I have a great opinion of her/him and I genuinely think s/he deserves that,

If they do include specific questions in their letter to you, then answer those, or explicitly say "I cannot answer question 3 because...".

There's a good chance that this professor's department's or university's policy is to send requests for letters to every recent student of a faculty member who's up for a promotion. Look for an official document called something like "appointment and tenure policy" on the department's website; if you can't find anything relevant there, go up to the university level or the "school" level and look some more. To give you an idea of what they're like, here's the appointment and tenure policy document for my university.

Don't second-guess whether you were enough this person's student for your opinion to be helpful. The promotion committee thinks so; that's what matters. (Was this person an official member of your thesis committee? If so that's plenty good enough for most departments.)

Do be honest about the depth and breadth of your interaction with them. For instance, you could lead off with "I have known Dr. Lastname for 3 years. I was never enrolled in any of their classes nor did I have very much interaction with their research group at $UNIVERSITY, but we regularly met over that time to discuss the research leading to my dissertation on $TOPIC." And then go on to talk about how they helped you with your dissertation.

Since you and the professor collaborated on things and you are happy about the result presumably, just describe the collaboration and what it meant for you. I think that in this particular university teaching and fostering students is valued. In some places it is valued more than research, actually.

So, if you think the professor has helped you along toward a career as an academic through your joint work (or otherwise) you can write about that.

The fact that you are "just" a post-doc may be an advantage. It will give variety to the professors promotion dossier. Others will contribute from different perspectives.

Just be honest.