A large dog or a big boar raccoon can tear hardware cloth or tear it loose from its connections if they can get a grip. That doesn't happen often, properly installed hardware is about as safe as you can get, but nothing is perfect.
I use electric netting, not a fence. The volt tester just says over 7,000, I'm not sure how high it actually goes. And it is pulsating, that is the key to safety. If it were a steady current you could get stuck but the pulsating current repels you. You can let it go and will.
I've had chickens pecking at the base of the netting and get shocked. They jump up and back two or three feet, squawk, and go about their business. Other animals do the same. it certainly shocks them but they don't get stuck and they leave.
For the electricity to work you need to complete the circuit. The fence is the positive charge, you have to have a ground for them to touch at the same time. With my electric netting my "ground" is the ground. I connect the ground wire to a rod driven into the ground. That means my hot wires cannot touch the soil. When a critter touches a hot wire and the ground at the same time they get a shock.
I don't know what your ground looks like. It could be the soil or it could be a ground wire.
A chicken's feathers, a bunny's fur, or a dog's fur will insulate it from the electricity, even if it touches a hot wire. But if a chicken touches the wire with its bare head, feet, comb, or wattles it will get a shock. A dog's tongue, nose, feet, and bare parts have to make contact before it gets shocked. Dogs tend to explore the wire or fencing with a nose or tongue before going through, yelp, and then run away. There are only a few parts of their bodies vulnerable yet they seem to always find that current. Their feet on the soil grounds them for my netting. There is something satisfying about a dog approaching the netting, sniffing it, yelping, and disappearing into the distance, never to come back. Actually they usually run a short distance to where they feel safe, turn and look at the fence, then goes away. Some dumb stubborn ones may need to be bitten twice to learn, but they do learn.
Small birds can perch on the netting or wire without a problem. They only touch the hot wire, not the ground, so they don't complete the circuit. Even if they did it would not injure them. The voltage is not what causes the harm, it is the amperage. The voltage just gets their attention. Yours too when you touch it and the ground.