Very few—if any—professions are stress-free. In the case of animal trainers, it's not normally your clients—the animals—that may cause you stress—it's their owners. People with problem pets tend to find their way to you only after a very long time trying to deal with the problem themselves (or working with a trainer who didn't really work out). Their expectations are high and they're expecting miracles—today.
If you're working in the entertainment industry, then actors, directors, lighting designers, and costumers expect trained animals to act like, well, people. To sit for hours, to speak only when spoken to you, etc. Animals in the entertainment business tend to be very smart and very attuned to their trainers—you. But they're not people (which is the reason, after all, that you decided to go into this business in the first place).
Sure, any job is stressful, but then again, animals are proven stress relievers, right?