Yep, we have the same system in the UK. In fact, the envelope looks almost exactly the same so they might even be printed by the same company.

You get two envelopes (one big, one small), a postal voting statement, and a ballot paper.

The actual ballot paper just has a list of options for you to put your X against; there’s no personally identifiable information on it. Once you’ve filled it out you seal it in the small envelope.

You then fill in the voting statement (it has your name and address on it so they can cross your name off as voted, and you sign it so they can check your signature matches the one on file) and both that and the sealed ballot go in the big envelope. That way your vote is still private because they check the vote is valid in one step and then add your ballot to a pile to be counted with the others in a second step, at which point it’s anonymous.

https://www.electoralcommission.org.uk/voting-and-elections/ways-vote/how-vote-post