Yes the clutches are similar, and made by the same manufacturers. However, clutches of this type come in a myriad of configurations. Diameter, belt width, belt type(v & poly-v), number of grooves, offset, bore diameter, bore type (tapered or straight).
Automotive A/C compressors generally use single groove or poly-v. The single groove ones are often fractional HP belts(more narrow than the A section belts used on TM units). The A/C compressors almost always use tapered bores.
TM units use pumps that have straight shafts, not tapered. CAT pumps use metric bores, i.e. 19 mm. Hypro pumps use fractional inch straight shafts. All pumps have a unique mounting, usually designed for one model number of mounting plate.
The bottom line is this. A typical Auto Parts store is going to stock clutches that fit automotive A/C units, and not the particular models that are going to fit your pump.
However, Auto Parts stores often buy their clutches from distributors of clutches that carry entire lines of those clutches. manufacturers, such as Ogura, make all manner of clutches, for industrial applications, as well as A/C units. But, be also aware that many of these clutches get private labeled before hitting distribution, and therefore the part numbers grow rather unwieldy for simple crossover.
This means that some auto parts stores are going to be able to order your clutch, or one that fits. it entirely depends on whether the store management is interested in such sales. You may find that all the guy behind the counter only wants to know is "what kind of car?". if so, he isn't going to be much help.
NAPA stores are renown for helping people in your situation. However, whether they are faster/cheaper varies wildly.
The bottom line is that TM Distributors compete for such business, and hopefully offer the best balance of service/availability/price, for the TM owner.
In my particular position, and experience, I always dealt directly with clutch manufacturers for purchasing, and never had to scrounge at the Auto Parts store level. So someone like Bob Vawter may be of greater help with part numbers for private labeled equivalents.
Good luck.