Should a neutral wire ever be connected to the neutral for the power meter?
I am not sure I understand the intent of your question.  The neutral wire is not metered.  It just passes through the meter box.  Old electric meters don't have a connection to it at all.

Electric meters monitor both hot legs and add up the kilowatt hours based on current flow through those.

 photo source.

The only remarkable thing I see in your photo is the unusually small number of circuits.  That makes sense if heating, air conditioning, hot water are provided by the landlord or commons and you don't have an electric range, laundry, etc.  Nothing but some lights and electronics.