The function which would allow that is Show variable which retrieves a variable value, while Set variable either creates a new variable or changes its value.
The problem here is an input list though named doesn’t create a variable of the same name making quite useless for your purpose.
A workaround is to create a new File>Properties in Custom Properties. It can be referenced through its name via Insert>Cross-reference, DocInformation, Type Custom (or the equivalent field).
However, there is no notion of closed list. When you modify the property with File>Properties, you can enter any text, which may bring in a value you did not expect.
There may be other workarounds. For that, edit your question to better describe the use of the choice in the list (e.g. to show a different version of the document, …).
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