They might have an office but they don't have a bank. Why not? Because operating a bank requires a huge amount of capital and a shit tonne of regulatory compliance. Literally 10s of millions in lawyers fees alone. Per year.
So no one bothers having more than 1 EU entity. Offices maybe, but not entities.
Starting in a week, the London entities that all those big banks set up are useless for offering services to the rest of the EU.
This is a big issue because its NOT enough to just get a PO box and register a company. You need to capitalise and regulators want to meet traders. You need to pass the local exams and comply with the local regulations.
Otherwise 101 foreign banks would not would have such a big London presence, they'd just run it from their home locations (Hong Kong, New York, Singapore).
For a long time, the UK has had a big advantage because the FCA were rapid and easy to satisfy. You could get a bank up and running here in just 2 years for less than 100m. Now you have to justify keeping that office when 80% of its business now needs to be done in Frankfurt, Paris, Dublin, Zurich or wherever.
This all seems like a lot of fuss and a bit anti business right?
The reason we all (America, Japan, EU, HK, Singapore, etc) do this is because its the only way to avoid major crises with huge knock-on effects. Capital requirements came in to avoid another sub prime bailout. The tests and training and licensing of individuals has grown up after 101 scandals, most recently the rate fixing on fx. Europe especially has learned that under regulation in finance is much more expensive than prudence. And prudence for complex, large, interconnected, opaque, big banks is pricey...