Welcome to the Wefugees Community and thank you for reaching out to us.
The platform and our experts are based in Germany. Therefore, we might not be able to give detailed answers or to share experience regarding the marriage process in Kosovo.
Do I understand correctly that you have a Finnish passport as Finnish citizen? If that is the case, I don't see a reason why authorities in Kosovo will not accept the passport as identity proof. I tried to make some research and found a website of the Embassy of Finland in Pristina which gives information on the marriage registration process in Finland once you married abroad.
However, I assume (and I also found explanations on a Swiss page here and an US website here) that you will need to provide additional documents as a birth certificate and a certificate of no impediment (showing that you are not already married).
I hope this helps and all the best,