Unknown Wire/Hose?

The small hose shouldn't just be hanging down from the battery, the bottom end should be going out through a grommet in the floor. If the battery ever vents, the sulfuric acid, hydrogen, or hydrogen sulfide goes out of the car instead of into the cabin.

The wire under the back seat looks like a previous owner had some kind of thing they wanted to wire in.

I can't ID the brown thingy off the top of my head, but the wiring diagram probably can:

Toyota Service Information and Where To Find It | PriusChat

 

That hose plugs into the side of the battery, under the positive terminal, it's a vent.
If there is no place to plug it in, you likely have an acid battery, which is incorrect for the Prius.
And not safe for you!

The brown connector goes to the left of where the fan connects. In the picture, bottom left corner,
you can see the white connector, that's where it goes. I don't believer the car will run without it.
The name of it and what it does seems to be hiding in my mind and I can't recall it.....

Someone will likely chime in though.

The red and black wires are aftermarket. Previous owner might have had an amp, or other powered item.
You can probably just remove it.

 

Previous owner likely put in a non-standard (12 volt) battery, without the vent function. With the battery being in the cabin compartment, it's much preferable to have an AGM (Absorbent Glass Mat) type of battery, complying with the spec's for dimensions, ratings and vent port.

Dealership parts department will have them or can order, though if it's like North America you best be sitting down when you get a price quote from them. There's likely third-party batteries availably as well, meet specs and a little cheaper.

 

There’s a few things definitely not as they should be. “Brown thingy” is meant to be clipped to metal tang on hybrid battery base, and various electrical connectors should be plugged in. Ditto for 12 volt battery vent tube.

I took a few pics of ours, attached:

 

Maybe some sort of tapered stopper with a center hole, that the vent hose could push onto?

 

It's supposed to be a tight fit. Look carefully, is that plastic part on part, or two parts?
You could wipe some silicone on the and press it in.