Learn Hardcore Latin: Pater idem dicit quod mater dicit. - Father says the same thing that mother says.

A very common first reaction is to assume it describes pater, as if it meant the same father, but that is not what is happening here. In this sentence, idem is the direct object of dicit, so it means the same thing.

So the structure is:

Pater = the father
idem = the same thing
dicit = says

So: Father says the same thing ...

Why is idem not a different form, if it means the same thing rather than the same father?

Because the form idem can represent more than one grammatical possibility.

Here it is best understood as neuter singular accusative, meaning the same thing. That is the form Latin uses because it is the object of dicit.

It just happens that idem is also the form for masculine nominative singular. So the form looks ambiguous by itself, but the sentence makes the function clear.

In other words:

pater is the subject
idem is the object

So idem does not agree with pater here.