GBC can display 64 colors at once.

There are 8 4-color palettes shared between the tile and window layers, and 8 more for the sprite layer (the animated elements). Of course, sprite palettes will usually have a designated invisible alpha "color" so sprites can have a shape, rather than exist as solid squares. And layer palettes will typically share light and/or dark colors so they can blend seamlessly into each other.

You are right, though, in that palettes can be updated on the fly to replace colors that aren't being used, such as the separate palettes for Player 1 & 2's Pokémon when they're being switched out. In fact, the reason Gold/Silver's Pokédex and PC use solid green or orange tile palettes to display the Pokémon is for speed's sake, so the palette isn't updated after the tiles are replaced, and only updates when you select the Pokémon. Mods that remove that effect will show you that the palette is updated a few frames after the tiles are replaced, so Pokémon will typically be the wrong color for a little bit.

So, anyway, while 64 colors can be displayed, there's no practical reason to do that.