In the original script the Terminator checks the Sarah Connor victims for a known scar on her ankle. Presumably Ginger would not have had this scar either, but then again, neither did our Sarah. In actuality, the scar was supposed to be from when the Terminator grabs Sarah's ankle in the hydraulic compressor. So even if the Terminator did kill the right Sarah, he wouldn't have found the scar.
Aside from this double-checking, it's likely that the Terminator doing kills with no name first would do a little research first to verify who he killed. I'm not sure if a Terminator would automatically assume that a resident there is automatically Sarah Connor, hence asking the others, "Sarah Connor?"
Also, chances are that the Terminator had a series of secondary objectives if it was indeed successful (or if it thought it was successful). Part of those may have been eliminating other threats to Skynet, other parts may have involved aiding research for Skynet in some fashion. I know it was meant to be a paradox, but presumably Skynet didn't know that unless it liked fulfilling the paradox rather than actually trying to kill Sarah Connor.