Yes, the UK (together with most other Anglo-Saxon-Celtic countries) doesn't issue residence cards to its residents — this is why we resort to providing organisations (such as banks, the local library) with multiple documents to prove our identity and residence: passport, driving licence, utility bills, bank statements, and so on.

Separately from that, the British terminologies for legal residence for non-citizens are confusing, and out of line with the terminologies used by most other countries, European and otherwise. "Leave to enter" causes particular confusion among non-native English speakers, "leave" meaning of course "permission", but for most users of English as a second language, "leave" means only "depart", the opposite of "enter". "Leave to remain" isn't much better. The UK has been working to simplify some of its terminology and passport stamps, with some success (I guess).

To the OP: the simplest thing will be for your wife to apply for a Schengen visa, which ought to be granted free of charge if you provide proof of your British citizenship. The application should be made, as noted above, to the Schengen country that is your main destination; if there is no main destination, it should be made to the country in the Schengen area that is your first point of entry to the Schengen area. Don't forget that the UK is not in the Schengen area (and nor is Ireland, although Switzerland and Norway are, even though they are not in the EU).

You will find it easiest to travel around the UK and contintental Europe using your British passport, although your US passport would be fine. Don't forget that you will ned your US passport to re-enter the USA. There is no problem with travelling with both passports.