'Tessera' or the plural 'tesserae' are Latin words that are not used in everyday English. Perhaps it's a token system used by this particular pub?

It depends on what the doorman means by tesserae. The term tesserae was mainly used in ancient Rome. Romans would give out tessaraes as tokens, or tickets for food or for entry into the theatre (just some examples) . When people ask for a tessarae now, they usually mean a ticket or an invitation of some sort.