Why is hot water coming out of every single faucet in my manufactured home? Water heater is brand new and just put in yesterday.

As the water heats and expands, it has nowhere to go but back out into the cold supply. You may need a thermal expansion tank or, if you already have one, it may have gotten waterlogged, i.e. the bladder that held the air had collapsed and there's no longer any air cushion in there, just water.

The new heater is probably larger or at least more efficient than the old one. Your piping system probably had a water hammer arrestor or just a simple vertical piece of pipe, closed off at the top end, which was preventing pipe vibration and rattling. If the old heater was smaller or not as efficient, the volume of the hammer arrestor was probably enough to also double as a small thermal expansion tank and now it's not. The arrestor itself might have gotten waterlogged and that needs to get fixed.

So, you can do several things:

Lower the temperature setting on the heater to the lowest acceptable temperature. The lower the temperature, the smaller the thermal expansion.

Drain both hot and cold pipes completely to restore the air bubble in the water hammer arrestor device if there is one installed. Since it could just be a piece of pipe hidden in a wall, I would do it anyway just in case.
If there's a thermal expansion tank installed, add pressurized air to restore the bladder or to create an air bubble that will act as a bladder but will need to be re-supplied every month or so because air dissolves in water. You may also opt for a new thermal expansion tank if you see water coming out of the pressure relief valve when you check the pressure. If you don't have a thermal expansion tank, you may need one.

If your water is supplied by a well pump, there may be more to it than only thermal expansion. We would need to know more about your water supply system then. Posting manufacturers and model numbers with questions always helps.