Sort of. Your explanation is basically correct, but one minor detail is off. Travel agents were issued generic ticket stock that had 10-digit stock control numbers. Then they were issued a plate for each airline on which they were authorized to issue tickets which had a 3 digit number on it (e.g. 016 for United). The agent would then use an imprinter (like an old credit card imprinter) to print the plate in front of the stock control number to issue the ticket, thus creating the full ticket number. Thus they had one plate for each airline, but just a generic ticket book.

Travel agents are still issued a series of stock control numbers, so each travel agent may have a very different number starting immediately after the 016. For normal travel agents, barring some fancy manipulation, the stock control numbers usually increment sequentially. For example, if I issued a United ticket that was 0160000000001, then immediately issued a Lufthansa ticket that ticket number would be 2200000000002. (I think that when some airlines sell directly they may try to be more random to prevent third parties from inferring data based on ticket numbers.)