"He's invited the wife and I" and other similar sentences are referred to as Toff's error. What is, precisely, the meaning of this term?
Toff's error is synonymous with the term hypercorrection. It refers to the erroneous use of a word form or pronunciation by a user who believes s/he is indeed using a correct form. And moreover, that this "correct" form is actually a correction of a common mistake.
Toff is BE slang for an upper class person.
"He's invited the wife and I" is wrong but is mistakenly phrased to sound very correct and therefore upper class. It should be "my wife and me" but people wrongly feel that "me" is wrong.
Of this construction, the distinguished authors of ‘The Cambridge Grammar of the English Language’ write that it
. . . is used by many highly educated people with social prestige in the community; it should be regarded as a variant Standard English form.
To those who say that me is required because that's what it would be if it occurred alone, they say:
But why should we simply assume that the grammatical rules for case assignment cannot differentiate between a coordinated and a non-coordinated pronoun? . . . The argument from analogy is illegitimate. Whether [the construction] is treated as correct Standard English or not . . . , it cannot be successfully argued to be incorrect simply by virtue of the analogy.
'What the old man does is always right' -- that's Toff's Error for you in short. Technically, it is an error of grammaticality, though in colloquial use, it is accepted and in fact, considered stylish.
If the wife weren't around for a moment, you'd understand the error of grammaticality: ? He's invited I [correct form: He's invited me]