Currently I am applying for PhDs in social anthropology in UK. I have multiple applications going on at the moment. One of my possible supervisors is directly working on the issue I proposed for my research, and she is also undertaking her fieldwork in the same city I proposed. At first I thought this was a big advantage but when I talked to her, she suggested me to make minor revisions in my proposed fieldwork since there should be "a gap" between the supervisor and the student. Now I am not sure how much of an overlap is too much. Is there anyone who had a similar experience?
While I don't have similar experience, as you describe it, yes, there is too much overlap here. You don't want to be in competition with your advisor for publications.
You do, however, have a somewhat unique opportunity to discuss how your research and theirs can complement one another, leading to a better overall understanding of the underlying issues.
I think, also, that it will take more than just doing the same research in a different location unless that location has fundamentally different characteristics, such as being in a different culture (for some questions).
Another issue for you at the moment is that this person is probably ahead of you on the research path for this question. You are therefore at risk even if you get a position elsewhere with this project as the basis. I suggest that you rethink it a bit, perhaps generalizing, perhaps modifying. Don't be chasing someone with more experience who is already ahead.
I understand from your question that you want to modify a project proposal to apply for a PhD position in the UK.
Ideally, you should be free to write a proposal that matches your expertise and interests, since you will be working on the project for at least 3 years. However, you also need to increase the chances of being admitted and maybe obtain a scholarship.
If your project has a significant overlap with that of your supervisor, your work might be seen as not novel and not "useful" to the department, because it is just a copy of someone else's work. This could also undermine the requirement of "independence from the supervisor", which is required in EU grants (MSCA and ERC), in case you decide to apply after the PhD. You mention your potential supervisor is working in the same city of fieldwork and on "the same issue", so I guess you are referring to the same research question. If this is the case, then it's important that you modify your research question into a novel one.
Let's assume your research question is "to study how antisemitic extremism is spreading through school-age people in the UK", and you will apply feminist ethnography methodology by observing and conducting semi-structured interviews in the local sports community in Bath, you could modify your research question by restricting the observed community (e.g., only a specific age group, only a specific gender) or expanding it (e.g., school age but also university students), changing the fieldwork cities (other UK cities plus a comparative study in another country), using a different methodology (postcolonial theory applied to ethnography; adding computational methods), or even changing the object of your study (not antisemitic, but xenophobic extremism during Brexit).
If your research question is not the same as that of your supervisor, then changing the fieldwork city might be sufficient. I hope this helps.
If I try to put myself into the possible supervisor's shoes, I imagine having already planned and started a research project on my own or with collaborators. I have a potentially pretty precise idea about how to do this and what to do with the results. There was some planning about resources and required manpower, and the decision was to do the project in this way, without taking on another PhD student for it.
I don't really see a place in this scenario for an incoming PhD student who wants to do the same thing. I wouldn't want to have somebody who has somewhat different ideas on the same topic which I'd have to somehow coordinate with my own (as a supervisor I'd be involved with the student project but I had made my decisions how to handle this project before the student was there), and obviously having exactly the same ideas wouldn't work either. So I imagine this to be rather awkward and I wouldn't be keen on it.
That's just for giving you an idea of what this person might think. I don't really have an answer to "how much overlap is too much", but I'd be happy to supervise a student who does something that has some clear connection to my research but that doesn't interfere with what I'm doing already. (I do realise that you don't necessary plan to "interfere", but this is how it'd feel to me; if I supervise a topic, I'd ideally like to help to make the work address this topic as well as possible - but this is what I'm trying to do with the ongoing research already. So then I'd at the same time have to try to give you my best thoughts on the topic but not those that I'm already using... confusing.)