The ONLY ID that is accepted for Americans in Europe is an actual passport.
I am unsure on the "actual passport" element of the statement, as I don't know if a photo or scan of the main identity page would suffice (though I believe it would and if the police were interested enough in you, they could always accompany you back to your hotel to give you an opportunity to produce the actual document).
My main comment is on the "ONLY ID" part (all caps? please). That statement would probably be accurate only for Americans who are not resident in an EU nation (and possibly not American military personnel stationed in the EU as long as they have their military ID with them). For those Americans residing in an EU nation, the resident permit card would suffice (such as in France, one's carte de séjour, which also may serve as a European Blue Card allowing one to legally work).
Now, with the recent reinstatement of passport checks for travel between EU nations, I would think it wise for even those who have a valid resident permit from one EU nation to take their passports with them when going from their home EU nation to another. We did on a recent trip to Ireland, for example.
A few years ago we didn't however, due to an forgetfulness, when we decided to drive our car from Lyon to spend a weekend in Geneva. We had grown accustomed to tooling around France with only our cartes de séjour.
As we approached the Swiss border crossing I saw officials waving cars over and I puckered up as soon as realized we had left our passports in Lyon, given that Switzerland is not part of the EU. With considerable dread, I pulled over as directed, expecting the worst. Thankfully, all they wanted was 40 CHF for the vignette. No passport check or anything. Whew.
We were very law-abiding visitors for the rest of that trip in Geneva. None of the usual French jaywalking no matter what other people were doing. Strictly German with respect to walking to a crosswalk and waiting for the light to change. We wanted to draw zero attention to ourselves.