Is it possible for elementalists to use something other than a hat as their 'shrine,' as you put it? Like a mantle or a sash or something, for example.
Has to go on the head, regardless of anything else about it. The nature of stuff that you wear on your head favors it being a hat, but it could, conceivably be any item of clothing… as long as you are willing to wear it like a hat.
Magic gets really set in its ways regarding traditions like this because it uses these otherwise random signifiers to tell magic users apart and serve them appropriately. The magic users who founded Elementalism used hats, and it just stuck.
Could an elementalist specialize in multiple elements by having multiple hat shrines?
Nope, they are stuck with one. The hat serves a couple different purposes, but it is not where the Elementalist’s specialty resides, so taking the hat off doesn’t take off the specialty, nor does it bestow it on another.
The hat’s first purpose is to announce the Elementalist’s intention to be an elemental to magic. So the hat communicates to magic what their chosen element is, and magic accepts or rejects this. (magic will reject elements that are too broad in scope, for example)
Once they have that all sorted out, the hat acts as an on/off switch. They can take the hat off and practice whatever their magical mechanisms are (power words, gestures, etc.,) without anything actually happening. The hat acts as a magical beacon that says “OH HEY LISTEN, MAGIC, DO ME A FAVOR WILL YA?”
It’s not absolutely off when the hats are off, since magic isn’t like a computer or electrical process. It’s more like a very distracted and scattered consciousness, so it can be pleaded with to ignore the fact the hat’s off just this once please it’s an emergency won’t happen again I swear.