In the US, any tub made past about 1900 or so has an overflow drain that prevents overfilling. The adoption of such drains not quite contemporaneous with the adoption of indoor plumbing, but very soon after.

It's the metal plate usually just below the faucet that looks like it might be useless. But it has an opening underneath to let the overflow water in. These are always sized to handle more water than the faucet can output.

This is basically a non-concern in anything but very, very old, unremodeled bathrooms.

The sinks do as well, but it is more obvious.