Not sure about “my old lady”, but over here in the UK, the expression “my old man” has become confusing, as it now can refer either to “my husband” or “my father”, which suggests that one usage is a recent import.
Not at all, the dual meaning has been around since at least the beginning of the 18th century. Here are the first two citations for old man, n. 1.b.:
b. colloq. A person’s father; a woman’s husband or male partner. Freq. with possessive adjective.
1673 DRYDEN Marriage a-la-Mode I. i. 5 My old man has already marry’d me; for he has agreed with another old man, as rich and as covetous as himself. 1707 C. CIBBER Comical Lovers V. 71 You must, and shall love me, and all that; for my old Man is coming up, and all that; and I am deses peré au dernier, and will not be disinherited.
Like a lot of such expressions, it is only confusing when taken out of context. Since the term is used familiarly, the listeners and readers can be presumed to know whether the husband or father is meant.
There is also the military sense of commanding officer, which the OED dates to the early 19th century (late 17th century and Dryden again for the more general sense of a supervisor). This one is often odd in that the commanding officer is sometimes younger than the speaker.