This is my recollection as well.
As far as I remember, potential employees were required to prove they had the legal right to work in the UK, which would most often be by showing a passport of any EU (or other qualifying) country.
As far as the (potential) employer was concerned, there was no difference between a UK passport and, say, a Polish passport. There was no need for employers to keep records of how many UK employees they had, how many Polish employees etc. (In fact, some people would probably argue it would be discriminatory to keep such info).
Other countries may have different systems*, such as ID cards, which allow them to collect and compile such info. I've never worked in another EU country, so don't have any experience to offer.
But EU rules around FoM definitely don't/didn't require such info to be collected.
ETA *different pre-existing systems