You use Context for that. Very powerful. Navigate to your custom field in the Admin - Issues - Custom fields. Then click on the ellipses (three dots) and choose Configure. There you want to create a new context. Choose which project the contexts applies to (and you can narrow it down to issue type(s). Save the context. Then click on edit options and there you can enter the values.
Do the same for the other project and you are good to go :)
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