It is still a public through street. There are plenty of streets that exist with a single or no property along them, and that doesn't make them service roads. By 'private' I mean a private-owned roadway intended for customers, employees, residents, etc. to access a building where usage as a public street may be limited or restricted, which Linney Street is not in this case. It can be used as a street to access another street. 

Another tag that can be used to imply this is just a narrow public street with a service road-like function is highway=service with service=alley. Alleys are more often publicly-accessible than typical highway=service ways.