This myth is probably especially common when people are thinking about massage for themselves, but some apply the same idea to therapy for their animals. (It's a myth either way, whatever the species.) The fact is, most massage benefits can and should be achieved without inflicting pain.
Does this mean all pain is bad? No. Some pain may, at times, be necessary to accomplish a worthwhile "release," or therapeutic shift. But this is more the exception than the rule, and the practitioner must be highly skilled and accurate to reduce risk of damage in the process. Yes, damage is possible with massage and bodywork. That's one reason those without advanced skills and experience should avoid moves that the animal (or human) responds to with a defensive pain reaction.
While many benefits can be accomplished with zero pain or discomfort, some therapeutic effects fall somewhere in between. You may already be familiar with the "hurts so good" sensation, where there's some discomfort, but at the same time the body is still wanting the work because it knows a positive process is happening. You may see the animal really "tune in" and focus. They may go through moments of agitation, but then return to the inward focus and eventually display a clear sign of release and relaxation. So although it's not always true, a better rule of thumb for massage would be... Pain? No gain!