How is a PhD student staying in their institution as a postdoc perceived by future employers?
As someone hiring post-docs and staff members, I am leery of someone doing a long (defined as ~ >6 month) postdoc where they were a PhD student, particularly with their same advisor. Why? To me, the purpose of a post-doc is to have a clear demonstration that the new PhD can move into an at least slightly different area and rapidly come up to speed and be productive on something new. Staying where you are just means finishing up loose ends, whether correct or not. If in the 6+ month range, you've got some explaining to do to me in the interview.
I'm not upset by a few-month "postdoc" with their PhD advisor, finishing up things, as they are looking for the next position to come open. But a post-doc has to be something new - it is the time to show off your new-found abilities to learn and make progress. Why? Because that is what you are going to have to do for the rest of you career. You aren't going to keep doing your PhD project for the next 30+ years.
For my postdocs, what does showing off look like? We hand them a project, and expect that within ~3 months there should be progress sufficient for starting to submit abstracts to conferences. And then they get another new project to get going, with abstracts going out on it 3 months or so later. The point is that in the first year there should be conference presentations and papers heading out the door on the new work at my institution. They should have an interview talk based on that work. That is what a postdoc should look like, and what I look for out of their postdoc when hiring new staff members. And that does not seem to happen for a postdoc staying at their PhD institution under their PhD advisor - it is just too easy to wrap things up, not start new, well defined projects that are different.
I would like to add that for the next 2-5 years (or more based on what happens) I will be much more flexible with this. Clearly the pandemic is not going to help anybody's career trajectory, and the constraints on jobs, moving, and everything else will make life for grad students and postdocs far more complicated than usual (and everyone else, too). Still, choose wisely - what will make you better at what you want to do?
This is probably more a function of what you do and how you present it when applying later than anything. If someone stays because they have nowhere else to go it is a bit negative, but your description can be stated as a positive thing.
You are likely taking your current research further than you could otherwise. It isn't just the publication count, but the significance of what you can produce.
In a new position (in some fields) you will have constraints on what you can do, depending on the PI. Not so much in mathematics, perhaps, but more in some other fields.
I'd suggest that you can make it work either way, but think about how you present your choices when you move on.
Publications
I feel that there is a higher chance for me to produce publications if I stay with my current advisor
One of the very best outcomes of a postdoc is publications. Academia is essentially built on publish or perish. All other things being equal, I would go where your chances of publication are the highest.
I would also explore options for collaborating with the PI overseeing the potential postdoc in the nearby city. Is there still an opportunity to broaden your network by doing some collaboration? Just a thought.
I have evaluated a few job candidates who have a postdoc. (I work in industry, not academia). I usually do not care where they did their postdoc. I care what they did. Generally speaking, I would much rather take a candidate with three publications with their old PhD advisor than someone with one publication at an institution different than their PhD granting institution.*
Network
One thing I will mention is the benefit of opening an entire new network of opportunities by going to a new institution for a postdoc. At my former PhD institution (call it U of X), most PhD grads end up in some low level academia position (community college, non-research teaching university, etc.) Networking was really tough at this university.
At U of X, I shared an office with a lady who had an opportunity to do a postdoc at U of X. She had published several papers with her advisor already and likely would have published many more. However, U of X has mediocre to poor faculty networking (a discussion for a different day. Long story short, when a department hires a bunch of their own PhDs as professors, it's not good for networking). She decided to leave to another institution for a postdoc. She only published one paper during her postdoc, it just happened to be a really good one. She now is a researcher at an Ivy League school. Her postdoc blew her networking wide open.
Be careful to examine where postdocs from both opportunities end up. Network matters a lot in getting your first "real" job. Do not compromise network for one or two extra publications.
*Do note that I would question a postdoc who did his or her BS+MS+PhD+Postdoc all at the same place. It might not be a deal breaker, but it would certainly make me wonder if there are some underlying issues.