There's no need to refrigerate used cooking oil or any cooking oil.
I've never refrigerated used cooking oil after I used it in my deep fryer.
After using my deep fryer with cooking oil I just allow the deep fryer and cooking oil to cool off and then put the lid on to the deep fryer to keep bugs and other things out of the cooking oil.
The cooking oil will then last for me up to 6 months before I need to change it out.
If you use the deep fryer for cooking food more often then you might want to or need to change the cooking oil more often.
But changing the cooking oil in the deep fryer every 6 months is good enough for me and I always just leave the cooking out out on the counter inside the deep fryer covered until I'm ready to use it again.