Can it be unfair when the same referee reviews a manuscript again at a different journal?

If you have concerns that the reviewer isn't being fair or just, you can always contact the editor and ask for a new reviewer. It is not entirely uncommon, I know of researchers that have a "blacklist" of people that they do not allow to review their works.

But consider that the reviewer might have a point. I know it is not your case, but I get really disappointed when I get the same paper twice and it is exactly the same text. Regardless if is the same process or not, a proper review would most likely improve the paper, so to see that not even the typos got fixed (in my case) is hard.

More to the point, how did you change your paper to make sure nobody else reading it would reach the same conclusions as this reviewer?

So, can it be that our common policy of giving multiple reviews of the same paper, especially when we don't propose any improvement to be done, is unfair to the authors as we can be the ones wrong?

I don't think it's unfair. Moreover, I have a feeling that if it was the positive review of your paper that you got twice from the same reviewer, you wouldn't come here to complain that this was unfair. This reminds me of the joke about a professor who had a student who failed the final exam in his course and had to retake it. On the second attempt, the professor gave the exact same final exam, and the student failed it again. He then went to the professor and said "professor, I think it's unfair that you failed me a second time." Incredulous, the professor says "but I gave you the same final exam and you still failed it!", to which the student replies "exactly - I already failed that exam so it was unfair of you to give the same one; you should have given me a different exam!"

Jokes aside, what is potentially unfair is to get a negative review from someone who clearly hasn't understood your paper or even bothered to try. Even then there is an argument to be made (that has been made here by various people in connection with past questions) that that means your paper is at fault for not making it easy enough for the referee to understand your results. In any case, whether it's unfair or not, the unfairness is the same whether that lazy/ignorant/mean-spirited referee reviewed your paper once, twice, or any other number of times.

And can I, as an author, do something about this situation?

Yes, you can. You can write the editor a polite email in which you give your best attempt at explaining why the reviewer's report doesn't make sense and should be ignored. You should have done that already after the first submission, but you can equally well do it now after receiving the same negative report the second time. You can also point out to the editor that this is the second time you're getting the same referee report, but personally I don't think that's a valid argument that should sway the editor's decision. My advice to you is to focus on the substantive issue of why the report is not a useful one, and not the number of submissions it was used in.

Yes, like almost everything in life, it's possible that it's unfair, e.g. for the reasons that you describe. Often, though, it's an advantage:

• It puts less pressure on the system in general, since fewer reviewers are required per paper. It's much quicker to get up to speed on a revised paper if you've read a previous version before.
• It's almost always possible to find something wrong with a paper. If it's sent to a new reviewer each time, they won't see the improvements being made and will simply identify new things that need changing. You'll get caught in an endless cycle of revisions.

And consider the alternative: if papers were sent to a different set of reviewers each time, authors would be more able to play the odds, and simply keep submitting until they happen to get a favourable set of reviewers. As the most suitable reviewers are used up, it gets sent to people with less expertise until the paper's (possible) flaws get overlooked. This is still unfair, just in a different way.

Finally, bear in mind that a reviewer cannot accept or reject a paper, that job lies with the editor. Therefore, the editor provides a check against possible rogue reviewers: if the reasons for suggesting rejection are unreasonable, the editor is at liberty to ignore them.