Your title question is a little different than your details because Bench Press has one other unique call out when thinking about barbell vs dumbbell and that is hands are at completely different angles. Therefor when you are using a barbell, you are working more muscles (like triceps) and dumbbells are doing a better job at targeting the chest more precisely.
Benefits of Barbell Bench Press
- You can accommodate a heavier load due to an easier set up (with jcups) and more stability
- Can be easier to track progress through wider range of weights
- Easier to spot and therefore safer to take on larger weights
Benefits of Dumbbell Bench Press
- Since each arm is independently holding weight, you are able to better identify and work on addressing any weak side
- DB Press allows for a larger range of motion which helps to target your pectoral muscles
- Because of the instability of a dumbbell, you need to control the descent more which creates more time under tension
Now, for the title question. It's too difficult to answer this vaguely. I think to a degree, most will have more stability challenges, but something like a dumbbell squat really isn't giving you that greater of a challenge. However, dumbbell squats have their own unique advantages, like altering center of gravity based on where the dumbbell is placed.
If you really want to focus primarily on stability, I would advise hanging your plates from your barbell with a resistance band. This can be done for a variety of exercises—press, bench, squat, etc.