This is an interesting question and I spent a while digging into it. I believe the short answer to your question is that choosing a party while registering to vote is pretty unique to the United States.
While some things compulsory voter registration and compulsory voting are not unique and there are a wide variety of voter registration techniques different countries (and states) use, I found very little to suggest that party registration is as common a component of that process in other countries as it is in the united states. In fact, using the "registering with a political party" semantics you use to search for information seemed to exclusively yield information specific to the United States. This may be because I'm searching in English and from a US IP address however I played with the phrasing quite a bit in order to get international results and still had no luck.
Also, as noted by some commenters, voter registration/party lists in the US are not quite as public as you might think. I found this great breakdown by state that shares each US state's usage restrictions and what they keep confidential, as well as links to each's process for obtaining the data. That being said, these look like small hurdles for any individual or group with significant resources and weight behind them so I think your "public" premise basically holds.
Back to your question though, I also realized there's another way to research this: open vs closed primaries. Because...
In a closed primary, only persons who are registered members of a political party may vote the ballot of that political party.
California: No party preference
So, based on that definition, assumably anywhere there are closed primaries then there's a much larger incentive for people to register for a party (so they can vote on the potentially larger pool of candidates before the election). There seems to be a lot more data on this, and here it seems pretty conclusive that closed primaries are not unique to the US...
Closed primaries happen in many European countries, while open primaries have so far only occurred in the socialist and social-democratic parties in Greece and Italy, whereas France's Socialist Party organised the first open primary in France in October 2011.
That being said, the party choice is a. not public and b. not part of initial voter registration in those countries as far as I could tell. So, I think my initial "short answer" above still holds unless someone can provide evidence to the contrary (in which case I'm happy to edit this answer). Also note that even in the US, we seem to be heading more towards open primaries and there are a number of groups advocating against and/or discussing the topic...