Backdrivable actuators know how much current are they sinking - and thus, how much force are they facing. So even the simplest two finger gripper can easily "know" that one object is squishy and the other is not. The response curves are different. Same can apply to arm actuators for weight estimation.
Now, what this doesn't actually get you is an estimate of how hot/cold an object is, and how moist an object is.
Where cameras come in is specifically the "guide a grasp" part. Force sensing alone can tell a robot "the grasp is failing". A camera pointed at the grasped object can tell "why".
I spent years of my professional life on these topics[1] and from my experience motor drive current usually isn't enough to narrow down the weight of an object in a typical picking task even to within a factor of 2 to detect when you've accidentally picked up two items from a bin instead of one.
The power of modern AI robotics is that, much like the biological mess it's trying to replicate, it doesn't care that much about absolute sensing.
Imperfect sensors fused by a high performance control model with a good foundation of world dynamics and self dynamics? Gets you impressively close to "perfect sensors" in practice. Being able to take a big bundle of noisy nonlinear ambiguous measurements, and implicitly reference them off each other and over time to get to a usable control loop is powerful.
It's the kind of power human nervous system wields to force the horrid mess that is the human body to perform.
For "detect when you've accidentally picked up two items from a bin instead of one", a wrist camera is a far more natural sensor than a bundle of actuators - especially for lightweight items. Wrist cameras are awesome. But even a bundle of actuators, if fused well enough, can often do it.
The problem is that designing algorithms to sensor fuse to this level by hand is pretty much a no-go. The kind of sensor wrangling hell I would only wish upon an enemy. Which gets us right back to general purpose robotics being an AI problem at its core.