Not just analog video. The broadcast industry uses 75 ohm coaxial cable for all signal lines, whether analog or digital, and lately, most RF lines (that aren't waveguide) exclusively. The exception being broadcast radio RF systems which is still mainly 50 ohm stuff. That being said, even the digital audio standard for radio and TV signals (AES audio) is carried on either 110 ohm balanced lines (originally), or more typically nowadays, on 75 ohm coaxial cable. Why coax? Because one can assemble connectors to the ends of the coax much more quickly than preparing a balanced connector. It makes a huge difference when building studios that can easily contain 5,000 or more point-to-point connections.
Why 50 ohms? Back when it was developed as a standard, it was the best trade off between power handling and loss; frequency response wasn't much of an issue with the relatively low frequency transmitters of the time. This is why radio stations still mainly use 50 ohm transmission lines.
Why 75 ohms? Back when it was developed as a standard, it was the best trade off between loss and power handling. Power handling capability wasn't much of an issue with video signals, and it was an easy impedance to transform to the typical 300-ohm balanced conductors used to connect to antennas. This is why TV stations (UHF stations) have mostly switched to 75 ohm transmission line. Even a quarter of a dB savings of transmission line loss can save thousands on the power bill over the course of a year.