Heck I can get all sorts of documents saying I own a firearm. In fact I currently have 8 of them that was given to me each time I purchased a firearm. One for my records and one for my insurance company. They were provided to me by the dealer, not the state. My CCP was issued by the state but it is not a picture ID.
IMO, the FOID, as you refer to it would not be an acceptable form of ID, especially if it does not have a picture on it and if it's a document that doesn't meet the description of allowable documents in your CIP policy.
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The opinions expressed are mine and they are not to be taken as legal advice.