You have not provided enough information for anyone to give you a specific answer, as the responses above clearly show. Which US state and which EU country are the first things you need to specify.

Then you need to specify what reason she has for wanting both. Her reasoning may or may not make sense but no one can tell you that if they don't know what her reasoning is to begin with. For example, if she plans to spend more than 90 days in an EU country which only allows driving on a foreign license for 90 days, then that would be a reason to want a license from her home country. But if she is only ever going to go there for less than 90 days at one time, there is no need to get a license for her home country and any advice as to how to get one would become pointless.

The more relevant information you provide, the more relevant responses can be. As a simple question, 'can she have both', may make sense to you but in fact there is only ond simple answer to that question, 'maybe'.