Can an actor use their stage name to open a checking account (and deposit checks made out to a stage name), or to get a driver's license, or to sign a notarized document, or to get a credit card? Or do they always have to use their real name in cases like that? If a stage name is acceptable in those cases, what prevents anyone from using one?
In the U.S., you cannot get a driver's license in a stage name and cannot use it to notarize a document. (Germany permits certain kinds of "nicknames" such as a stage name or a name taken upon admission to the Roman Catholic clergy in addition to a birth name on certain forms of official identiciation and documentation, but makes it much harder to officially change your name entirely.)
In the case of a checking account or credit card, in the U.S., the bank needs your actual name as it needs to match your social security number which the bank also needs. But, the bank can put a different name on a check or on a credit card as a convenience user name, as a doing business as name if the dba is registered in accordance with state law, or as an entity account with the stage name treated as a unincorporated entity (much like a general partnership) with its own taxpayer ID number. As long as the name on the account can be tied to the name of a natural or legal person that has their own taxpayer ID, it can be done. But, otherwise, it would be a tax evasion tool and is prohibited.
The tenor of the question implies Witness Protection ID 44583292 is interested in the US. I am not aware of any law telling a US notary whether or not a document that gives the signer's stage name can be notarized. Obtaining an official ID in a stage name would be difficult, but notaries don't have to limit themselves to official IDs. Every state allows notaries to rely on identification by one or two credible witnesses. Every state except California allows notaries to notarize for a person who the notary knows personally.