Animal species have male and female specimens, and so do plants.

Not strictly true. Some animals, like snails & aphids, are hermaphroditic some
of the time.
On the other hand, most of the flowering plants have male & female elements in
the same flower or on the same plant.

Do plants and animals have a common ancestor that already had specimens of
male and female gender?

Probably not. The male & female elements were carried on the same organism, but
I will leave the experts to answer that one.

Does sexual terminology mean something completely different when applied to
plants as opposed to animals?

Not really. In animals, the female produces eggs, ova. The male produces sperm
cells which are carried to the eggs, either externally in the lower animals or
internally in the higher animals.
In flowering plants, the female part of the flower contains an ovary, which
contains little eggs, called ovules. The male part of the flower, or if
separate, the male flowers, called staminate, produce pollen, which essentially
contains sperm cells. Through pollination, the pollen cells manage to get to
the ovules through many different mechanisms, sometimes wind, sometimes
insects, sometimes self-pollination, etc. The terminology & mechanics are
different in plants, but the basic process is the same.
Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)