Based purely on aerial imagery, I'm gonna say that this is, in fact, an unnamed service drive.

The service drive immediately to the north, however: apparently THAT'S Rush Street. This is based on the Bing Maps Aerial that shows a path going towards the railroad tracks at the same location as the other Rush Street, the USGS Topographic Map showing Rush Street as one continuous road across the railroad tracks, county parcel data, and official county road maps labeling it as Rush Street (see: [https://www.loraincountyengineer.gov/engineer/pdfs/2022/Lorain-County-Roadmap.pdf]).

Unsure if it's signed, though.