Can I own more than one copy of the same set?

Yes. If you bought two of the same starter set, you can track both as separate copies in My Sets.

Just add the set again — each time you do, you get a fresh owned copy with its own member pieces, its own completion count, and its own condition, photos, and notes. Your second copy won't disturb the first, and it won't try to reuse the pieces you already logged.

This is handy for sealed-vs-opened duplicates, or one to run and one to keep mint.

For the basics of adding one, see How do I add a set I own?