Last night, I was running Hell's Rebels for my usual group. Without offering any spoilers, they were fighting a pair of creatures with pounce, four attacks that could apply both bleed and poison, and a poison cloud attack that happened as a free action after a pounce (provided that at least one attack hit).
One of them was pounced on, and hit with all four attacks. They saved against the poison for half of those attacks, then failed against the cloud, for a total of 3 failures. The section on poison in the Core Rulebook states that poison stacks, unlike other afflictions, but also clarifies that it stacks by extending duration and raising the save DC. Are these the only stacking effects, or does the damage of the poison get applied for each application of a poison that a person fails to save against? This is important because it's the difference between a person with 8 Wisdom taking 4 points of Wisdom damage (bad) or taking 10 points of Wisdom damage (death/catatonia).
Here is an official Piazo Explanation of Poison that will answer your question.
The quick answer is that the damage doesn't increase, but the saves and duration's do.
That is per type of poison though, so the claw and breath weapon might be totally separate depending on the exact nature of the creature in question.
In some instances, this is much more dangerous, as while the character might not take enough ability damage to kill or incapacitate them immediately, with enough stacks the DC can reach impossible heights such that neutralize poison and the like fail to stop the affliction from running its course.
I happen to like this approach. In fact, if I had my way I'd make heal work like neutralize poison with respect to afflictions rather than simply removing them--it's already an extremely powerful spell, and I like afflictions remaining relevant at higher levels without needing extra rules to say "sorry, only wish cast under a new moon on the second Wealday of the month will save you now"