Hi Tom,
When I read your post my first reaction was - children don’t need anything until they are 16 yrs old - however I mentioned it to my Spanish wife who was in the NIE office the other day with a client and she said there were some people in there getting NIEs for children BUT she is unsure of their nationality and any other details. I have put some links below but to be honest I think you need to check with your local town hall (responsible for schools) or if you have one the Officina for Extranjeros.
I think the card for families you mention is for Non EU…...
N.I.E. (EX14) / RESIDENCIA (EX16)
Application Forms and the
Regulations Governing EU citizens in Spain document.
http://extranjeros.mtas.es/
For forms - Click: Modelos de Solictudes
Or direct link
http://extranjeros.mtas.es/es/general/procedimientos_Solicitudes_index.html
For the Regulations document (from Home page)
Click UK flag & R?gimen de los ciudadanos Comunitarios en Espa?a
Let us know how this works out and what region you are in…....