I love these noob questions. They're disconcerting and embarrassing. Like, where do babies come from? The simple answer will let your inquirer confused. The complete version with the "sordid" details will let them horrified.
According to
this article, we do not use coax with the exact impedance of the air because that is not convenient from the standpoint of power and voltage. Waveguides have their restrictions too.
If you leave a transmission line open, you will have some propagation, but the impedance mismatch will make part of the radio wave be reflected, causing all kinds of nasty problems, some of them catastrophic in the case of power signals.
Antennas match the line with the air by simply propagating all or almost all the signal that is fed to them. So no signal is reflected. To do this several tricks are employed. In resonant dipoles, the standing wave will induce currents and voltages that will match that of the cable.
Feeders of parabolic antennas behave much like a speaker horn, by gradually changing the impedance of the guide.
The design of an antenna defines its bandwidth, directivity, polarization, and many other parameters, some of them legally required.
As for the antenna gain, the reference is an ideal isotropic antenna. It radiates power uniformly in all directions. If your antenna concentrates power in a specific direction, you will have a "gain" in
that direction.
So parabolic antennas have high gain, but they are very directive. Dipoles have low gain but they propagate in all directions perpendicular to the axis of the dipole.