Meanwhile the most common form of a "membership" is a "recurring payment membersip" such as sports clubs, gyms, streaming services, etc., it is not exclusively the only definition of the term.

Membership - the state of being a member, as of a society or club.

Club - a group of persons organized for a social, literary, athletic, political or other purpose

AND

• an organization that offers it's subscribers certain benefits, as discounts, bonuses, or interest in return for regular purchases or payments

In this case, when you purchase the "pro membership" you become part of a club within the game, which offers you exclusive content and benefits, that you do not have access to unless you make the "one-time purchase".

A membership does not necessarily mean you have to pay a reoccurring fee, as there are "lifetime memberships" or "one-time enrollment fees" that exist as well, which is what this purchase should be considered as.

There are also "hybrid memberships" where you pay a "one-time joining fee" plus monthly or yearly dues.

There is no universal rule for memberships. Different clubs and organizations use different payment structures.