The property was likely taken either taken by prescription or inverse condenmnation (if the city built the pathway). However, you need to figure out who built the pathway and why. This could have been built by the original property owner as a "neighborly accomodation" which would defeat prescriptive use claim. I would find out who paid for it, who maintains it and look to see if there is any record as to why it was put in. Depending on what you want to do these can be fact intensive cases. However, what ever you do don't tear out the stairway unless you are clear that you have the authority to do so.
I would suggest that you ask the governmental entity involved what they think the stairway is - if they built it - if they want it - what is the background, etc. My main concern would be your liability related to the use of the stairway - if it is going to stay in place you shouldn't have to be liable. Also, it would be nice to know if the stairway could be re-built in another location if that location needlessly encumbers other uses of your property.