2.4GHZ US Wireless Device in EU
The manufacturer specifically says that different bands are used in Europe compared to the US. 



If the band is occupied, now or in the future, you could interfere with someone else's signal or your signal could be interfered with. In many places it can be illegal to use non-approved devices, which in practice means that it could be confiscated at customs or if you interfere with someone else, it could cause issues. 

Can you get away with it? It's hard to verify their story or whether it's likely to result in interference (worst-case they're lying to prevent the type of arbitrage you want to do). 

The US bands used are 2404.0 MHz to 2480.0 MHz, according to their FCC submission. 



Here's the European spectrum allocation in that range: