Do main panels have to be mounted on a piece of plywood? Can it be mounted in between two studs then insulating fiberglass behind it?
If in the US, the language is more general; fundamentally, the panel must be solidly supported, and not prone to fall off the wall as time passes. I personally prefer to have 5/8" type X drywall over the actual support when using a wooden support, for additional fire resistance. It's not required, but as the comments imply, your local "Authority Having Jurisdiction" (AHJ) is what really matters. That is your local electrical or building inspector; however, you can exceed their requirements so long as you actually meet them. That is where I'd class my affectation for firecode drywall on top of wood .vs. exposed wood.
The wall can be insulated, but the service entrance wire should pass though as directly as possible, and preferably in conduit.
From 2011 NEC:
Having met a few things mounted this way, I can attest that wooden plugs in masonry rot, shrink, or both and do come loose, thus the specific exclusion.
Incidently, many panels can be mounted in the wall cavity between studs, though that can complicate the actual wiring in some ways. Perhaps that was what you were asking in the second question - it seemed like you were asking if a plywood mount could be placed that way. If it is your question, the answer is "Yes, if the box is designed for it and the wiring can be run in the space then available." It will reduce your wall insulation at that spot considerably. Many boxes will still need a solid support behind them, between the studs, to accommodate their mounting screw hole placement.
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In the UK, modern "consumer units" do not need to be mounted on a board. In most UK houses that means they are directly fixed to the inside of an external wall that in most cases is a plastered surface over construction blocks or brick.
You can get flush mount consumer units that are installed into the wall itself.
In previous centuries I believe it was normal to mount fuseboxes on a wooden board. There's probably still a lot of houses with that arrangement.