There are 75Ohm BNC and also mini BNC 75Ohm connectors.
As for why 75 Ohm, someone in the distant past thought it was a good idea, and it was different enough from the RF guys. Usually your video output stages will be matched to 75 Ohm impedance.
For analog video, the difference between 50 & 75 ohm connectors is small enough to be ignored.
The TV Studio & transmitter sites I worked at used 50 ohm connectors as standard, as they are the most commonly available, & (a less often considered point), are more rugged than 75 ohm connectors, because the PTFE part extends further into the actual connecting part.
They come in sizes to fit all the common 75 ohm coax.
When the Studio started piping digital video signals around, they changed to "true 75 ohm BNCs".
I'm a bit dubious whether it was necessary, as digital video is a pretty rugged format.
Anyway, that Studio is now gone, replaced with a Housing development!
By the way, as to worry about electrical noise getting into the video, using ordinary coax, I earlier worked at a site with a "high level modulated" TV transmitter.
To achieve this end, the Tx produced several kW of video signal, which then grid modulated the RF final.
You could hear the HF noise anywhere around the building on a HF radio, but we used (real) RG 59 to feed our (standard level) video around, with no problems with noise interference from this high level signal.
And before you object "But it's the same signal, how would you know?",
(1) We often tested equipment with different signals sourced elsewhere.
(2) Phase difference/ time delays.