How should I handle reviewing a paper that is very similar to something I am working on?
I am going to answer based on an implicit assumption I feel much of your question hinges on:


  One of the problems is, they did not mention that part in the abstract, therefore I have accepted.


This sounds like had you known that their work is so similar to yours, you would have rejected the review.

However, I would argue that that cannot possibly be in the interest of the review system.

Yes, you could argue for a conflict of interest, but it is the conflict of interest that exists generally, based on the simple fact that other people beside you do research in the same or similar topics as you. It is not a conflict of interest on a personal level (e.g. you being part of the team whose paper is being reviewed).

Instead, it is a conflict of interest that, in theory, should quite often exist in the review process - for reviews to be meaningful, reviewers need to be familiar with the topic to some extent, and thus, a similarity of the research topics between authors and reviewers is hardly avoidable.

I see your concerns about what the authors of the paper under review might be thinking, though you might have to brush them aside for the time being. However, what you can do is draw the benefit from being the reviewer and thus having early access to a manuscript:


While your method may be very similar, it is most probably not exactly the same. When describing your method, try and insert a few sentences that focus particularly on a few aspects that you do differently from them.
For future publications you might write on similar topics, keep in mind that there may be a similar publication by those authors around that you can cite.


This has advantages for both sides:


You increase the chances to include a relevant citation that you might otherwise miss (and thus the authors' chances of being cited by you are increased).
Even though you cannot yet cite the work of the other authors, your clear description of some different aspects serves as a way to "preemptively" establish the differences between your work and theirs.
If they happen to publish a revised version of their paper after your work has appeared, you have already provided them with quite a straightforward foundation for describing their differences to your work.




In the comments, the concern has been voiced that even without the assumption of plagiarism, the authors of the other paper might still feel by rejecting their work, you intentionally stalled their progress for long enough so you could publish your own very similar work first, and now it's their problem to care about the prior art.

However, there are some aspects of good scientific practice that speak against that line of reasoning:


You do not provide an unjustified judgement, you provide a review. If you choose to reject the paper, you have the opportunity to describe objectively and in detail why you deem the work in its current state not ready for publication.
The other authors have not yet published their work in a properly citeable fashion, but that still just means you have plausible deniability on your side when you are asked whether you knew about the other work. Scientific integrity still demands that you consider the existing prior work, published or not, which is exactly what you are doing by specifically describing some aspects that you do differently than the reviewed paper, as described above.