Why would a mammalian vector plasmid require an antibacterial resistance gene?
I think the assumption you are making is that the ampicillin resistance gene would save a purpose in the mammalian cells.
It does not.

As you said, it is only use during the cloning process to select the bacteria to replicate and produce more of it.

There is a resistance gene on the vector you posted which affect mammalian cells: neomycin. In this case, once transfected into mammalian cells, you could select only the cells with the plasmid by adding neomycin the cell culture medium.

Note: neomycin selection can take some time (4-5 days) and higher dose than for example puromycin. Hope this helps!