Recently, a collaborator of mine received a paper to referee from a journal, and he proposed me that we review the paper together. When he submitted the report, he indicated to the journal that he reviewed the manuscript with me. Am I allowed to write in my CV that I am a reviewer for this journal?
You can list reviewer activities on your CV, especially in an early stage of your career, see e.g. https://academia.stackexchange.com/a/3543/68222.
The situation you describe is different. You weren't one of the official reviewers of the paper. As stated by @Dirk, this situation is uncommon and might not be compliant to the journal policies. Because of that, I suggest to not mention this specific "review" on your CV.