If i undertand correctly, you can rename all radio buttons that unison with the same exact name in the buttons and forms panel.
Yes we already use the same name for all gender radio buttons within Indesign. Using the same name is enough for text fields, checkboxes,... to work in unison.
AFAIK only radio buttons need the option set explicitly, otherwise although the name AND the radio button choices are the same, they are treated as separate fields.
Unfortunately, the availability of button and form field functions are not completely parallel between InDesign and Acrobat. This is an example of an Acrobat form field interdependency feature, most (all?) of which cannot be set from Indesign.