What is worse is when people engrave their social security numbers or drivers license on goods. If I put my SS number on something, and it is stolen, the number can be used to open accounts, or used by an illegal alien to get a job. My brother was one of those who put his SS number on his things. Years ago his house was burglarized, and everything was stolen. Not one item was ever recovered. He died a few years later, and in the probate process, debts came up from around the country from people who had used his SS number. There were also 18 people using his SS number at their jobs.
As for license numbers, there are more than 50 issuers of licenses, and even if an officer can tell by the number what state or territory the license comes from, it is not always easy to get a name to go with the number.
For thieves who file false reports using stolen numbers, I never saw this during my 10 years n law enforcment. The thief needs to put the time and place of the theft, and generally show some type of evidence of a crime. He also needs to assume that the owner of the item doesn't have a receipt for it, or have it listed on an insurance policy, or even have old photos of the item, proving ownership. Also, stealing a camera, gun, or television is a minor misdemeanor, punishable with a citation, fine, and restitution. Filing a false burglary report is a serious felony, which can result in arrest, a fine, and imprisonment.
I have never heard of a case in which a false report was filed using a stolen serial number. The odds of such a trick working are less than winning te lottery.