I don�??t have a complete answer for you but I do have some ideas.
Bear with me for a sec as I rephrase your post as I�??m interpreting it, let me know if I got this right. I assume the road data you are referring to is street center line data in a line format. Both the City and County keep separate sets of data, the county�??s is obviously on a macro scale. You work for one of the agency�??s (City or County), and you want to make use of the data from the other agency for analysis purposes.
In the small city I work for we have come across a similar issue in that the City & County have documented different things with different purposes i.e. the County performs emergency response routing therefore they have fields for speed limits & addresses. The City has fields for bike lane, General plan, street sweeping, trash pickup, etc. All we do is scalp data from their dataset we are interested in via a street name join, we dump what ever data of theirs we don�??t want, including their line geometry as we like ours better :-).
I�??ve only used the intersect tool when I had a polygon in the mix, I�??ve never used it with just lines, I�??m not sure if it will work being that your lines don�??t necessarily share geometry (it seems like ideally you want to eliminate one set of lilnes and just transfer the data over to the other). It only takes a few minutes to give it a try (depending on your file size). I think a spatial join might be what you are looking for, though I�??ve never used it for lines that intersect. I�??m not sure how it will work for you. Let me know what you find that works 🙂
Someone else probably knows better than me but aside from a spatial join I don�??t know of a way to match up data with table lacking shared field data, I�??m not sure you could create a common identifier field unless it was done by hand. You may consider performing a join between fields with the most similarity ( i.e. street names) then check and see how much hand cleanup work is required for the street names that don�??t match, this might be a good way to transfer the other agency�??s ID field to your data as a starting point.
Whatever you do ideally you can structure the task so as to not have to duplicate each time the other agency updates their data. You may consider contacting the other agency and seeing if you can both agree to maintain a common ID field.
Hopefully something in all of my rambling helped out a bit.