There are lots of opinions here .. but looking carefully at the equations which determine the impedance of free space, these are related to c. (C0 to be correct)
Importantly so is the wavelength of a specific frequency, and from that the physical antenna length required to give good radiation.
So .. when you choose the antenna dimension to be a multiple or sub multiple of the wavelength, it is this that matches to the impedance of free space..
Importantly, the antenna changes the mode of transmission from a waveguide, coax, twin or other, to the ejection of electrons into free space, i.e. no longer constrained by a waveguide unless and until some group of these electrons meet a new resonant waveguide in their path.
Matching the radiator element to its reference plane, as Richard Harrison mentions, is a separate issue. We generated an AC electrical signal with reference to a ground or balanced opposite. The antenna is part of that AC circuit, and the AC pulses must feel the same medium (impedance) all the way to the point where the energy is released into free space.
There are various simulation videos on youtube showing that it is at the peak gradient of the antenna waveform when the electrons are accelerated enough to escape their waveguide, and it is the exposed resonant element which facilitates this change of propagation mode.
So in summary .. the choice of antenna length is related to signal wavelength, which is related to C, which also determines the impedance of free space. And this length is what matches to free space (if one must refer to matching here).
The matching of feed line to a balanced or ground referenced radiator is entirely separate.
It is standard AC and transmission line theory, of which a radiator is matched to its opposite or reference conductor.
BR
Tom

Importantly
On Monday, 24 July 2017 at 16:53:44 UTC+2, J.B. Wood wrote:
On 07/21/2017 05:32 PM, wrote:
The reason we don't match the antennas to the impedance of free-space is very simple, it all has to do with Industry standards, manufacturing, and application. If you increase the impedance of the coax to match a 377Ω antenna and transceiver, the design of the coax would demand that you increase the dielectric diameter ≈ 230 times in order to preserve the current capacity of the center conductor.

Hello, and I think you're confusing the intrinsic impedance of free
space (377 ohms) with the feedpoint impedance of an antenna, which
depends on frequency, physical dimensions, and construction materials of
the antenna. It's the antenna feedpoint impedance that we match our RF
source to, not the free space impedance. Sincerely, and 73's from N4GGO,

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J. B. Wood e-mail: