European citizen travelling to another European country without valid passport

For EU citizens, there is a small risk of minor inconveniences.

• EU citizens are required to be able to present a passport or acceptable substitute at the German border upon request by the authorities, as set down in §8 (1) 1. FreizügG/EU.
• The probability of having to show papers at an internal Schengen border is very low. It might go up if one matches the "typical profile" of a drug buyer or seller. Anyway, he OP's partner crossed the border. (Other readers of this answer might be less lucky.)
• EU citizens are required to own a passport or acceptable substitute while in Germany and to present it upon request by the authorities. They are not required to carry it with them. If he gets into a traffic accident or the like, the absence of papers will complicate things, just as it does for a German citizen who does not carry an identity card with him/her.

Intentional or negligent failure to own papers may be punished with a fine, but I would not worry about that very much.

In Schengen, ID cards appear to be viable substitutes for passports as travel documents, so the Dutch Identiteitskaart should suffice, if he happens to have it on him.

I thought that I did the same, travelling from Greece to Denmark (only having a European driver's licence and not my ID). According to this link, it's not a travel document, so you can't use it. At the same time, you shouldn't be asked for any travel documents, but you never know.

In my case, I did have my ID with me, but the airline told me that they wouldn't accept me on the flight if I had forgotten it (nothing to do with border checks).

As previously said, you do not actually need a passport, although it's good to have one, and especially for Germany-Netherlands it's unlikely that you will encounter trouble.

It's just the responsible thing to have travel documents with you.