Both "club" and "organization" are examples of collective nouns. In standard American English we would always use singular verb agreement with both "club" and "organization."
However, in British English the rules are somewhat different, and it is not uncommon to see plural verb agreement with collective nouns like this.
Although it is not uncommon, it is not universal. In fact as stated in the Wikipedia article referenced, often the distinction is in the context of the usage.
In British English, it is generally accepted that collective nouns can
take either singular or plural verb forms depending on the context and
the metonymic shift that it implies. For example, "the team is in the
dressing room" (formal agreement) refers to the team as an ensemble,
whilst "the team are fighting among themselves" (notional agreement)
refers to the team as individuals.
In the context of the example you provided, i.e. formation of a club, formal agreement appears to be the most commonly used. A search of the BNC shows only one example of "club were formed" whereas "club was formed" occurs nine times.
Thank you to @snailboat for helping to improve my answer.