So, bearing in mind that I;'ve never seized a gun from a lawful carrier for "safety", cops can document stops of course and many would record the serial # in the case they knew it. I just can say it's never come up that I've had to seize a gun from a CCWer (apart from 1 dui), or an OCer.
If the intel is "non criminal", then generally it cannot be shared outside of the police agency. I know many agencies write reports on 'verbal' domestics, which are noncriminal, but they actually take case reports, so not sure if they share those, but the rule is that LEA's cannot share noncriminal intel.
There ARE intel groups where people share info that are NOT law enforcement agencies and thus they do share intel. Some LEOs are members of these NONlaw enforcement intel groups.
If a guy seized your gun and recorded the serial # I assume he would run it to check for stolen. THAT would leave a record in the computer aided dispatch records. I would assume, to answer your question, that many LEO's would write up a field contact report in such circ's and that report could have the serial # in it, if the officer chose to include it. There is nothing preventing him from doing so, but again if the contact was noncriminal in nature, it cannot be shared outside the agency.
Contrary to a lot of TV portrayals, the databases we have aren't the way they appear on TV, like one could easily search for that data, because different agencies often have proprietary report writing programs etc. Otoh, some databases (like Linx) in SOME cases allow text searches of narratives etc. such that that info could be shared outside the agency reporting.
I can just state that no agency I have worked for (3) has any sort of database like you are referring to.