I think they don't overbook as much as in the US. They try to avoid bumping off passengers, because the cost of doing it is much higher. They can't just give you an upgrade ticket. They have to reimburse all occurring cost and they have to pay an additional 250-600 EUR compensation depending on the length of the flight.[1]
[1]: https://europa.eu/youreurope/citizens/travel/passenger-right...
I've seen airlines offering compensation worth well more than these minimums, however. Anecdotally it seems fairly common to get hundreds of dollars for a relatively short domestic flight.
Also, the ones they do bump off are the standbys and reduced fare, which is why you don't often hear about fully paying passengers being forced to take another flight - and in any case, if they're still over capacity, the promises of free upgrades and money mostly result in enough volunteers to prevent that from happening.
I'm not sure about any differences in US regulations vs EU regulations for this issue though.