Use of the superlative when only two items are present
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.
No, she isn't right. The younger of two children is also the youngest. (Of course, both are correct, so if she prefers "younger" in her own letter, that's perfectly reasonable.) Indeed, there's a joke about referring to the only member of some category with a superlative (the joke being that the sole member is both the most X of its type and the least X of its type).
Some prescriptive style guides insist that the superlative can only be used when comparing three or more things, and that the comparative must be used when comparing only two things. If you are writing in a very formal situation, you may want to follow this rule.
However, in practice people often use the superlative for comparisons between only two items. This use of the superlative is completely acceptable in speech and may be used in writing for most audiences.
I had never heard of this rule before reading your question! It's quite fascinating ...
To my ears, the superlative sounds awkward when used to pick out one member of a group which is generally understood, or explicitly given, to consist of two members. For example, all of the following sound odd to me:
- my oldest parent
- his left nostril is smallest
- the youngest twin
- the smartest of the two
The oddity of the above suggests that there is something to what your mother says. I would say that if it is known that there are exactly two members of the relevant group, it is more natural to use a comparative.
On a side note, quite a number of languages pattern the same way, so it is clearly not an idiosyncrasy of prescriptive English; rather, it might (might!) be a universal component of the meaning of superlatives.