The default way this is done on Android is having multiple bitmaps rendered at different pixel sizes for different DPIs (similar to the way mipmaps work in 3D texturing, if you're familiar with those) and then the OS chooses the one closest to the display DPI and resamples it as needed to get the desired final size based on the actual display DPI.

Eg:

You have some icon, you might provide bitmap versions of it at: 8x8, 32x32, 64x64, 128x128, 256x256 (the exact sizes depend upon the desired size of the image when displayed combined with what DPI ranges you want to support well).

Based on your layout, the final pixel size occupied by that icon control might be 200x200 on a high res tablet, the OS will choose the 256x256 bitmap and downsample it slightly to fit 200x200. The same icon area might be 60x60 on a phone, so it'll chose the 64x64 and downsample that one a bit to fit 60x60.

I haven't done much iOS programming, but from what I understand it works pretty much the same except the sizes are more fixed at 1x/2x/4x because there is less DPI variation across iOS devices.

This way has downsides (makes the assets larger since you have multiple copies of each one), but generally works pretty well in practice and actually works better than vectors for some things (though vectors can scale up and down at will, if they aren't heavily 'hinted' you can easily lose important details that you don't want to lose at small sizes, with pre-rendered bitmaps you can adjust for this ahead of time).