Mrs H, from what you have written, your DD was British at birth, she can never choose not to be, unless she renounces her British citizenship. Current cost of renouncing her nationality would be £229. A lack of a British passport DOES NOT mean you are not or no longer a British citizen, you don't need a British passport as a British citizen.
If there is a German nationality rule that at 18 years old for example, a child with German and OTHER nationality must choose and German is chosen, it means having to pay the renunciation of British nationality fee. I have no ideas at all about German nationality rules, only British ones. There are no such rules regarding British citizenship.
Gabsid, I take it you are German and living in the UK and that your child was born in the UK too? It's complicated with a child born of an EEA national in 2005 and in the UK. If you or dad were given indefinite leave before birth, your DS would have been a British citizen automatically at birth and you apply for a British passport directly if you wish and pay the £46 if applying inside the UK. If you or dad didn't have indefinite leave before birth in 2005, you can then "register" DS as a British citizen if you register registered for indefinite leave (but if here 5 years after 29.4.2006 requirement for the EEA parent to get indefinite leave was lifted and after 5 years residence it became automatic). But as you say you left, I assume left the UK? Oh I'm not sure then, if you made it to 5 years in the UK before leaving and it was from 30.4.2006, I think you keep your permanent residence for life, even if you leave the UK, so you might be able to still "register" your DS as a British citizen. Well if you mean you didn't leave the UK and are still here, then no complication about leaving the UK! As for your DD born in 2008, well if born in the UK to you or dad being 5 years or more previously living in the UK too, your DD will automatically be a British citizen, whether wanted or not, again having a British passport does not confer British citizenship, it's the circumstances which do. Yes a British passport can be handy for a working holiday in Oz or Canada etc.