Aircraft tires are indeed standardized by size and specification and with applications across models.
There is certainly a long list of available sizes and variations, just as with automotive tires. When needed for a new aircraft type design, these existing tire models are available to be incorporated into that design.
There are certainly many aircraft types that have necessitated special tire specifications through the history of aviation development, especially with the more esoteric military aircraft. Of course, this is how that extensive list of available tires came to be.
Aircraft tires are dimensioned by several measurements (as depicted in the images below):
- Outside Diameter (A),
- Cross Sectional Width (B),
- and Inner (Rim) Diameter (C).
These dimensions will be noted as AxB-C. In some cases the A is left out for a B-C notation (see examples in the table of "Type III Nylon Tube Type Tires" below).
Air Hawk tires are one brand that I handle frequently; their tire products give a good representation of the generic sizes and specifications available. Follow the above link for examples from their online catalog.
See also this excerpt from their Aircraft Tires Specifications sheet showing examples of common dimensions and plys:
Goodyear has an exhaustive databook covering technical specifications for their commercial and military aircraft tires.
See that here: The Aircraft Tire DataBook
Specific data on the proper tire sizes for different aircraft main and
auxiliary/nose gear, speed ratings, nominal inflation pressures,
dimensions, and other data needed by users to obtain the maximum
service from their aircraft tires are presented in this booklet.