As a reviewer, is it inappropriate to make unsolicited comments on another reviewer's comments?

It is appropriate. You're acting in good faith after all, the editor is likely to be interested in what you say, and you're not harming anyone.

This is however the kind of thing to put in the confidential comments to editor box. The authors don't need to know of friction between reviewers.

From my point of view, it is a task of the editor to make sure that you as a referee only get those parts of the communication which are meant for you. Since the response of the authors to another referee's comments is now available to you (intended or not), it is quite impossible to not factor it into your own consideration.

Therefore, I would point out to the editor that, strictly speaking, this thread of the comments is not originally yours -- but now that you have got insight to those points, you can just add a remark saying that in your professional opinion, you deem the authors' response convincing.

This makes clear to the editor that there might have been a mistake in providing that part of the communication to you, and the editor can decide for themselves, whether your stated opinion should be taken into account for the ultimate editor's decision, or not.