The point of peer review is anonymity but since most authors upload their papers to arxiv long before the paper is published (at least for math, stats, computer science etc), the reviewers should see who wrote the script by simple googling right? Will this have an impact on the review process? That is, a famous scholar's article will more likely be accepted to top journal given similar quality.
It is in fact often easy for a reviewer to find out who an author is, unless the author is quite new to publishing and has not uploaded the paper to arXiv.
Sometimes it is hard not to know even without a search since the author may be building on their previous work so the list of references will be a giveaway even if the the text doesn't refer to "my previous work in [reference]".
That said, refraining from such a search is implicit in the unwritten contract between the reviewer and the editor.
I hope and suspect most reviewers honor that contract.
This is extremely field-dependent. I have never heard of double-blind reviews in mathematics: never got to referee a paper that had the author's name redacted, never seen a paper where the author made a point of hiding if cited work is indeed theirs (e.g. "X did , Y did " instead of "X did , we did "). But at least some psychology/sociology journals I know maintain the policy of double-blind review, and I read papers in these fields that were written in third person, as in the example, apparently to conceal themselves. I have my doubts if this can work at all - even if the bibliography doesn't give the author straight away, the reviewer is often expected to check if the prior work in the field is well represented and discussed, which leads to queries and googling, which again can give the reviewer a pretty clear picture who they are reading.
And to comment on the opening of your question: the main point of anonymity of the reviewer is always that they are protected from retaliation of a rejected author, and since the author I am reading may well be my own superior (how many experts on foo of bar are there?), I am glad that I enjoy that protection. This is fundamental for the confidence in the process, regardless of your field.
But the other way round, it is not as clear cut: we all accept end expect the reviewers to give their expert opinion. But in some subcommunities we have agreed that said expert opinion can or should take into account who the author of the paper is, of course your mileage may wary on whether you like it or not. You describe one hypothetical often given (but not as often observed in nature) as an example of corrupting effects of this situation. As a sometimes disgruntled author, completely sure that I would have been accepted for publication if only I had more recognition, I must admit that stories of top journals publishing long series of sub-par papers of famous mathematicians are exceedingly rare, and if happen, tend to end badly for the editors, authors, and sometimes reviewers. I think that for the most part we do believe this market regulates itself, or at least we don't think that the prevailing single-blind in mathematics is the main problem in urgent need to be corrected. But again, your mileage may vary.