In my area of research double blind reviews are used to minimize unconscious bias. The earlier a reviewer knows the authors, the more she may be biased.
The goal of anonymity should thus only be to remove the obvious hints. Hide author names. Instead of "our previous paper " write "the paper ", etc. not making major changes. If you think the authors could improve in this regard, you could mention it as 'minor comment' similar to how you would mention typos.
In my area anonymity is a fickle thing. In particular, for papers the following make it night impossible:
1) Experts: Commonly there are only few people who are likely to publish on a given topic and if I am asked to review a paper I am an expert and probably know them.
2) Follow up work: They build strongly on previous work and seem to know all the ins and outs. Moreover, structure, writing style, idiosyncracies of this particular work are identical. Guess what.
3) Preprints: There is a good chance that you google the title and find the paper on arXiv.
4) Writing style: The most devious one because it pervades even reviews, which should be anonymous. Do they write British or American English? Do they make certain grammatical mistakes typical for non-natives of a certain background? Do they overuse CERTAIN words?
If you combine these, there is really no chance to anonymize papers in many areas.
In fact, 1) and 4) suffice for an expert to make pretty good guesses about stuff that definitely should be anonymous, like reviews.