If I am not wrong, I once encountered a British word used (in slang I suppose) to describe a person who boasts about going out, knowing or dating famous people. Does it exist? In this case, is it common?
Are you thinking of name-dropping (MW)?
the studied but seemingly casual mention of prominent persons as associates done to impress others
Name-dropping is the verb form, the noun is name-dropper.
The Term "Groupie" is often used in this case.
While, most commonly used in reference to a band/musical artist, it can also be used in context with any famous person or group.