Medium format is a width (60mm), just like 35mm is a width of the film in the camera.
In practice a medium format image was 56mm wide because there was an unusable area of 4mm and 35mm image was 24mm wide because it had a big unusable area with sprockets on both sides.
The length of the image was completely up to what camera you had.
One complication of that comparison of film dimensions is that the 56mm is used as the long dimension of the frame in most medium format film cameras (56x42mm for "645" and 56x56 with "6x6") whereas 24mm is the short dimension of standard 35mm format frame, 36x24mm.
In Europe, it is common to refer to that last format as "36x24mm", and I am sure that photographers who can handle exposure information involving several numbers (shutter speed, aperture ratio and ISO speed) could handle two numbers in describing a format. And we did it with 6x4.5, 6x6, 6x7, 5"x4", 10"x8", and often still do with print sizes.
On the other hand, with most cameras using formats in a small range of shapes from 4:3 to 3:2 (an 11% difference), the single measure of diagonal in mm should be good enough for most practical purposes.
Some examples:
16mm for the 13.2x8.8mm of 1" format sensors
22mm for the 17.3x13mm of recent 4/3" sensors
22.5mm for the 18x13.5mm of the "official 4/3" video sensor format
27-28mm for the various "APS-C" digital formats, ranging from about 22.5x15mm to 24x16mm
30mm for the 25.1x16.7mm of the APS-C film format
42mm for the 36x24mm of "35mm" still camera format
55mm for 44x33mm
67mm for 53.7x40.4mm, the current largest sensor for "digital medium format"
70mm for 56x42mm, the full "645" film format