The FAA has bilateral agreements with other countries and the EU for certification[1]. I read recently but can't quickly find in this that the FAA trusts EASA's certs for Airbus flying in the US.
The Ethiopian agency also trusts the FAA and EASA for this information[2]
So, it's not really tricky here, we can blame the FAA for the crash of a US manufactured plane in a foreign land (if the problem was the FAA's certification process).
[1] - https://www.faa.gov/aircraft/air_cert/international/bilatera...
[2] - https://web.archive.org/web/20180215175924/http://www.ecaa.g...
So, EASA can’t really abandon blame as easily as they abandon responsibility. If they allow a flawed process to continue then that’s on them.
However, in this case these agreements still give EASA veto authority about what aircraft are allowed. Which is why EASA was able to ground these airplanes before the FAA did.
Even larger, well-resourced geopolitical entities like the EU rely heavily on the FAA to do its job for Boeing aircraft, much as the FAA relies on the EASA to do its job for Airbus aircraft. It's not pure rubber stamping, but it's close enough that things like this can pretty easily happen if a manufacturer's home agency falls asleep at the switch.
An example of the FAA's influence: Airworthiness regulations are so closely coordinated that special numbering is used by Canada, Europe, and Japan to match the numbering of FAA airworthiness regulations for easy cross-referencing. The FAA is also delegated air traffic control responsibility for much of the world's international airspace by the ICAO.
It is basically impossible to fly a commercial aircraft on Earth without being under the – possibly very heavy – influence of US civil aviation authorities, and it would take a great many resources and serious, concerted political will by many countries to change that.
So, no they where not operated in the same fashion. It’s still been grounded in the US for good reasons, but you can’t take individual regulations in isolation.