Learn English – Is it correct to say “the oldest child” when you have only two children

It certainly used to be a rule, but there are grounds for ignoring it. This would be strongest where you were uncertain of the number of children (not the case here). But you can certainly argue that in logic, there is an oldest of 1, 2, or more children. So, your mother is almost certainly correct about what she was taught. Whether that stricture still has to be followed is considerably more debatable.

I think your colleague may be getting at the idea that than is a conjunction, not a preposition. This is a long-running dispute. Those who argue that than is a conjunction, claim that it is followed by a clause in which the verb phrase can be elided. When that happens, the only word left must be the subject of the clause. Since her cannot be a subject, it would be considered wrong. Other people consider than to be a preposition. If that's the case, then they reason, a pronoun following it would be an object, so her would be correct. Both forms are used, but the than she form is formal in register while the than her is neutral.