Engine moves with transmission shift?

Yep that's engine torque flexing mountst doesn't seem too too bad.

 

Something to keep in mind, if you’re doing something like mounting an oil catch can.

 

Worn engine mounts? Check lower engine mount underneath.

 

I suggest you replace that. The rubber doesn't compress much so the motor motion you see now is from those cracks and the gap on the bottom (if that isn't just a shadow) closing and opening. When that horizontal crack close to the nut goes all the way across and that block of rubber works its way out there will a lot more motion than you see now.

Some people repair these things by filling the gaps with various rubber compounds. It needs to cure for a day or two before use or the new stuff will just extrude out. Replacing it would be a better fix, but filling with rubber compounds is much less expensive.

 

What thing? The motor mount? The rubber in the motor mount? For the latter search for "fill motor mount".

This isn't a recommendation but this page gives a pretty good description of how to fill a motor mount:

https://www.suspension.com/diymmi

Looks like the cure time is longer than I remembered. Caution, many people try filling, find the ride is then too rough, and
end up buying the OEM ones anyway.

OEM mount part numbers (don't know if all of these are still available, probably?):

https://www.amayama.com/en/genuine-catalogs/epc/toyota-usa/prius/NHW20L/8936/engine/1107

There are also aftermarket motor mounts, see for instance:

2007 TOYOTA PRIUS 1.5L L4 ELECTRIC/GAS Motor Mount | Shop Now at RockAuto