Main service electric panel has no neutral wire - both neutral and ground wires on same bar - sub panel question
Up until 2011 the National Electrical Code allowed a "3 wire" method of connecting a sub-panel. This means you have the two "hot" wires and a neutral running to the sub-panel as opposed to the "4 wire" method where you add a separate equipment ground wire.

So, when your house was built, it was perfectly legal to use the 3 wire method. There is no problem with this unless you connect a metal pathway, such as a water pipe, from one building to the other. This allows neutral current to flow on this parallel path. This could be potentially lethal to someone working on the pipe.

As to the main panel not having a neutral: it does. If you look again you will see the two hot wires and a bare conductor. There is no ground wire on the service drop. The bare conductor is the neutral. This is allowed on the service drop, but from that point on all neutral conductors must be insulated. Ground wires are always allowed to be bare.

So, in short, there is no problem with your system unless you run a metallic system to the sub-panel building. This would create a problem.