That's actually not how it works. For example, a home 120V circuit's amperage is determined by the wire size. When you wire up a home circuit you could have a 1A device attached to it or a 20A device attached to it. You don't know. Typically, if the device attached to the circuit needs it's own protection due to smaller sized wiring within it, that device will have it's own fuse protecting it.
So you're saying every lamp or other device in your home that plugs into a 120 volt receptacle has it's own fuse/circuit breaker at or in the device? I know you didn't mean that.
Those branch circuits, if protected by a 15 amp breaker for instance, are limited by that breaker. Nothing over 15 amps can operate on that circuit, whether it's one device or a combination of devices. If you plug something in that draws 20 amps, the breaker will trip. So, you DO know what will function on that circuit whether it draws 1 amp or 20 amps. The breaker determines that.
A branch circuit feeding standard convenience receptacles in your house is always limited to 15 amps. The same applies to all your lighting circuits, 15 amp breaker. If you need more amperage, you don't increase the wire or breaker size for these types of circuits, you install additional circuits.
When installing any device, the first thing you do is determine its amperage draw. Then you figure the amperage of the breaker needed and then match the wiring to that amperage. This is how it's done. You don't figure the wire size first. That is the last part you figure.
Only specific circuits for appliances and motor loads can be 20 amp or higher. In your kitchen, you are only allowed two receptacles on a 20 amp circuit and that wire has to be 12 gauge. A window A/C unit that draws under 20 amps should be the only thing on that circuit and it should be 12 gauge wire on a 20 amp breaker.
So the breaker protects against an overload and indeed protects the wires, if the wires are sized correctly.