more likely ...a valid licence recognised by the country over which you are flying.
No. MJ was perfectly correct - the exact wording is, "a current pilot license issued by the country in which the aircraft is operated may be used". In other words, to operate a US-registered aircraft in the UK, the pilot must hold either an FAA licence or a UK
issued licence. An EASA licence issued in any other EU member state will not do.
However, returning to the OP, the statement in question is nonsense. An EASA licence issued in any member state allows the holder to fly an EASA aircraft registered in any EU member state in any (other) EU member state. My local club has been operating an EC registered C172 for ages, both for training and SFH, with no issues at all.