For proof of ownership I scan a copy of the receipt and keep a copy of it on my hard drive. The paper receipt gets placed in the safe. I do keep a database with serial numbers and lots of other information, this file is encrypted and password protected, but can be accessed using my cell phone if needed.
In the fake buffer tube I keep a printed copy of the BATFE letter that states that my Slide Fire stock is perfectly legal. No one has ever wanted to see the letter, but shooting it sometimes means lots of questions.
I placed a discreet p-touch label on all of my firearms with my name and cell phone number, but the database with the serial number is the option that I would use to prove ownership if a dispute ever occurred. The paper original will hold up in court if it ever got to that point, but the real reason I keep the original information is if I ever need to file an insurance claim. I have pictures of each of my firearms to back up the condition of the guns.
One reason to carry some sort of way to prove ownership of your gun is if there are two otherwise identical guns at the range and someone gets confused about which one is theirs. Of course it is always possible that there is no confusion, rather the "other guy" is trying to switch his worn out gun for my gun that is in better shape.