I haven't been to Europe in 8-9 years, so I don't know what the situation is there. But in the US, I can't remember the last time anybody at an airport ever checked to make sure that my checked-luggage tag matched my claim check/ticket. Not to give anyone any ideas, but it seems like a good opportunity for thieves to hang out in baggage claim areas and swipe bags, if nobody is checking.
In all my travels, I've only been checked once at the airport in Atlanta back in 1995; never been checked in Europe. The bigger threat is a fellow passenger taking your bag accidentally...happens all the time. But most of the baggage carousels at European airports are before the customs checkpoint, so it's mostly impossible for bags to be stolen outright. Airports have so many security cameras in place that any thief would be identified very quickly, so it's a big deterrent.
I have had baggage tags checked recently, but only in Asia. In India you can't get into the airport until 2-3 hours before the flight, only with a valid ticket, and all luggage goes through X-ray on entry to the airport. I've even had to ID checked bags on the runway before they were loaded there. Different kind of security... This is another reason to travel with cheap-looking luggage - bet any thieves would go for the pricey stuff, not my 10-year old backpack.
I don't have that problem. Using Rick's advice, I carry my luggage on. No waiting for the carrosel, no chance of it being stolen. Works for me.
You can't get to the baggage claim area unless you have arrived there by plane. People can't just go back there and hang around. A thief would have to be flying too. Sounds a bit expensive to me, plus yes, there are cameras everywhere. I saw open to the public baggage carrousels in Columbus, Ohio at their airport and was appalled, but it isn't like that over here.
Here in Denver, passport, baggage claim for international flights, and customs are all in a secure area. I don't think unauthorized people can enter it - only those coming off international flights. However, the domestic baggage carousels are open.
Los Angeles usually (but not always) checks baggage tags. Also happened to me in Miami
I've only ever had my checked-luggage tags looked at in China, never in Europe or the US.