Will cars produced be of the same model, same brand in the Equal Money System

What I see is that we will have only one brand: the Equality brand. It will offer the best cars we're able to produce. That means for instance that it will be long lasting and easy to maintain and repair. As many parts as possible will be re-useable, and the rest will be easy to dismantle and returned as resource material.

There could exist a form of competition between different factories to see which one can come up with the best solutions, technical/environmental/other, which will be shared equally, one brand would be most efficient and effective, and would depict the Equality principle nicely.

Inventions to make it even better will be decided on from a point of necessity and the harm it causes (in the physical world to living organisms) to implement the innovation. Retrieving resources from the Earth for instance may cause harm to bugs/animals living there.

There will be different models for different purposes. But not for personal taste. Everything will be aligned to the physical. So there will be cars to transport humans, to transport humans and goods like a pickup truck, for goods only, for any specific physical purpose we will make specific cars/vehicles, for instance for long or short distance rides. There might be a choice of chairs to accommodate the specs of one's body to the optimum, or a design that's easy adjustable for different body types. 

The cars will not be a personal possession, most likely cars will be available for communal usage. As this is more efficient. There might be cars for very long/short people. Though there might be people who like to have/share a specific vehicle for fun riding, then again will all points be taken in consideration if this isn't abusive towards other life forms.