They are simply a collector or hobbyist.
From: http://www.beermatmania.com/ "Tegestology is a Latin term defined as the practice of collecting beermats or coasters, with practitioners known as tegestologists"
Beer mats are sometimes more commonly known as beverage coasters. They are used to protect the surface of a table from being damaged by drinks, typically as a result of condensation.
Tegestologists collect beermats as a hobby.
A person who collects shells is known as a conchologist.
A numismatist is someone who collects coins and/or medals a numismatist
someone who collects umbrellas
A catapultist.
Memorabilian.
hoarder!
Someone who collects beer bottles.
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dumb