"Ladies like our cats."

Es ist unwahrscheinlich, dass das gemeint war und es ist auch kein vollständiger Satz, aber theoretisch ist das natürlich möglich.

I like your logic, but you are taking logic too far. You are forgetting something very small, but which is nonetheless REALLY important in the English language. Also you are missing the meaning of the sentence. ‘Ladies like cats’ makes sense in English. Even though nobody uses the word ladies much any more. Anyway, leaving aside the fact that it is an old fashioned sentence, it is also a generalisation and a crazy sentence. ‘women like cats’. Most women I know do not like cats at all. Anyway, pushing aside these prejudices and distractions, you can never substitute the sentence ‘ladies like cats for ‘Damen wie unsere Katzen..’. However, what you can do is to say: Ladies, like our cats, love drinking milk.’ Notice what has happened here? You have now finally put in the missing COMMAS! They are small, but wichtig. Also you have to continue the sentence. You can’t just stop and say ‘Ladies, like our cats..’