Possessing "The Power of Thor"TM and holding Mjölnir or Stormbreaker appears to bestow a range of abilities on the wielder. This includes the ability to create a Thor-style battlesuit. We see this in both Thor: Ragnarok and several times in Thor: Love and Thunder when both Jane and Thor put on Thor suits.
Being clothed by the Power of ThorTM is definitely one that requires an intentional decision on the part of the wielder of said power. Thor only gets his battlesuit on when he wants to, not simply when he holds Stormbreaker or Mjölnir. Presumably Cap'n America could have had his very own Thor suit if he'd wanted one.