Does it make sense to ask for a better job title (e.g. senior
developer instead of developer) when negotiating with my current
Sometimes. Job titles have some effect when dealing with clients, when dealing with others in the organisation, and when dealing with possible future employers. If you want any of those people to think that you're a "senior developer" rather than a "developer", then there's a potential benefit in changing your job title.
Will improving the job title improve my perceived status in the
In your company, where people currently don't have formal job titles, I doubt it. But it's down to the personality of the people you work with. Consider whoever sits at the desk next to you. If they were given the job title "senior developer" tomorrow, would you be impressed and defer to their newly-increased status? Would you assume that they're in charge of all future projects in which they work with mere "developers" like you? Or do you actually already have ideas about your colleagues' seniority, that a mere change of title won't alter?
Is this something more than just a small personal win?
If it's just the title then it's personal. If the management are going to back it up, by actually giving you a different role from the less-senior developers, then that's another matter, but that could happen with or without a change of title. Just negotiating for a different title isn't going to make them give you a different role.
If you're already acting as a senior developer, that is to say the mass of mere "developers" do already defer to your experience and skill, and perhaps you're paid at the top end already, then having different job titles to reflect different roles is more than just a personal thing. But for the organisation to reap any benefits they need more detailed job titles to be available for everyone. The fact they haven't done this suggests they don't see any big benefits for your organisation.
Just doing it for you wouldn't deliver the benefits, and might cause serious problems, unless you clearly are the most senior of all the developers and therefore changing just your title makes some kind of sense. If that person at the desk next to you is clearly junior to you, and showed up one day with a new job title "senior developer" and yours was still "developer", I suspect that wouldn't make you happy. Your employer probably won't want to give you a title that's misleading in the context of the titles of your colleagues.