For the US: Yes, the communist party issued membership cards. You can find a bit more about them here, which unfortunately is not a scholarly resource. Looking at a Report of the Committee on Un-American Activities, you can actually see that the "cards" were also used for bookkeeping, i.e. whether members paid their dues. This is, for what I can tell, certainly common in the Western world. Here is an example from the German social democrats in 1923. It contained some information on the member, some information on duties of members, and space to glue on stamps that serve as a receipt of paid membership fees. Here's a similar thing from the USSR in 1989

Another hint for the prevalence is that the german word, "Parteibuch", was first added to the standard German dictionary in 1934, but I should add that "Parteibuch" also can refer to party membership. Also, the German Social Democrats still hand out party membership booklets, but most parties and other organizations now switched over to credit card-sized membership documents that no longer serve a bookkeeping function.

I'd also like to add here that the use of little booklets is not limited to political organizations. The Swiss army used a "Dienstbüchlein" to log the service days done by individuals. Up until very recently, they still did so as a back-up in case the computer system would ever delete all data. More about this could be found in Beat Nann's Das Dienstbüchlein, but I don't know if you can get that outside of Switzerland.

There is also a second function of documents identifying you as a party or union member. The Slate article cites evidence (that unfortunately is no longer online) that some membership cards gave you a discount at merchants as early as 1927. If you've ever been part of a union, you may find that familiar, though these days (at least in Europe) it tends to be cheaper holidays, insurances or gasoline.