Sorry but, for UK citizens (travelling from the UK) it's a passport or nothing.
Travel within the European Union has always been possible using a national ID card, instead of a passport, but that doesn't help Brits (because we don't have national ID cards).
Further, under the Schengen Agreement, all border formalities have been abandoned across (almost) the whole of the EU, together with a few additional countries. As far as immigration authorities are concerned, anyone can travel freely between, say, France and Greece, or Latvia and Spain, or Italy and Lithuania. (That applies just as well to American, Australians and Argentinians as it does to EU citizens). The rule is that once you're in any 'Schengen' country you can freely travel to any other. The only ID that's required is that needed by the airline to meet their security requirements. Many European airlines will accept almost any form of ID for such purposes.
But here's the problem for your daughter:
There are 27 countries in the EU. 25 of them are already full signatories to the Schengen agreement, or will shortly become so. (As stated above, there are also a few non-EU countries within the Schengen group). One country (ireland) wants to sign up to the agreement but can't do so because of the obstinacy of the neighbour with whom they share an open land border. Yes, it's the UK that's the only EU country which refuses to sign up to Schengen.
Our politicians are too scared of the headlines that the Daily Mail and the Daily Express would print if we tried to join the Schengen group. For that reason, your daughter needs a passport. (If she was a Brit living in France or Germany she could travel to Spain or Greece without one. But, because she's starting from outside the Schengen territories, she can't).
Chris