Indeed, it’s meaningless to compare party “membership” in the US to being a real, card-carrying, dues-paying member of an organization in another country.
European parties can probably count on a higher level of active participation, but when it comes time to vote or donate, party membership definitely matters.
Party meetings in European political parties often involve checking you off against a membership register that in many cases is secret (not shared with the government in any way), and often e.g. requires you to be up to date with membership dues to have access to the meetings. Membership subjects you to rules, including behavioural, and commitment to a shared aim, restrictions on support for other parties, and the like. You can be subject to disciplinary hearings, and can be expelled. For many parties e.g. publicly expressing support for another party can be an expulsion ground. Some parties will do things like check your social media in advance of granting membership. Many have rules that requires a waiting time from you have been a member of another party before you can be admitted, etc.
You would not get more than a tiny fraction of the people who are registered Republicans or Democrats to subject themselves to that.
At the same time many of these parties certainly reach far outside their membership for donations, and none but the very smallest rely only on their members for votes.
It feels like they're comparing different things though. In the US, they use voter registration / party identification as "membership", while they use actual membership in other countries. My grandmother always used to vote for the Social Democrats in Germany, certainly identified with them, but was never a member. She wouldn't show up as a party member here, but she would if she did the same in the US. It seems "true believers" would be the better metric to compare to the US party affiliation.
Also, it's amazing how bad that website is built. Thanks for the bread crumbs to follow.