I've never paddle boarded with a dog, but I have paddle boarded a bit and agree with the above posters that you'll want to get a feel for it first. I just started this summer, so I'm practiced, but still don't have too much experience paddleboarding.
The more weight you add to the board, the tippier it becomes. I don't have a problem with it on my own, but I've played around on it with two people before and inevitably someone (or more often both of us) end up in the water. Granted, a second human is 120+ pounds and Jackson is little, so I don't know that that analogy is particularly insightful. Boards come in different sizes of course. I'm smaller than the average person that uses the board that we bought, so it's a bit less wobbly for me. On the flip size, we've had individuals who are heavier get on the board and it's a bit more of a workout for them just to keep it balanced.
That said, depending on the board, they are pretty stable, but I do recall just a few weeks ago making a move to adjust my weight on the board. I moved one foot and somehow ended up caught off balance enough to almost fall off and have to catch myself on my hands and knees.
I wouldn't bring a dog on a paddleboard unless you knew that they dog wouldn't panic if it ended up in the water. I think that a water-loving dog would be thrilled. One like Boo would be a wreck
Kayaks are certainly more stable than paddleboards. Still more of a risk of ending up in the water than on a boat, but not too much. I've never felt at risk of tipping a kayak except when trying to climb into it from a raised dock. Kneeling or sitting on a paddleboard adds to the stable feeling, but still not to the extent of a kayak in my opinion.
That said, I really enjoy paddleboarding and I wish I had a dog that would like it too (even if we were to fall off now and again).