Learn English – Are Titles Proper Nouns

Your daughter's textbook is not quite accurate.

You capitalize such nouns when they act as names, as the ordinary form of address. "Mommy" is used that way in your second example.

But in your first example, sister is not a name, as its use with a determiner makes clear: it is a common noun denoting a relationship.

If the determiner is omitted, however, the noun is treated as a name, and it is proper to capitalize:

Ask Sister. She's fond of playing on it.

That sort of use of sister is rare, although not unknown; as Damien H says, Sister, capitalized, is usually the title (of reference and address) for a Catholic nun.

Generally a proper noun uniquely identifies one person or thing. As such, different rules apply: it does not require an article, and cannot form a plural.

Cheese is uncountable, but we use the word cheeses. We are actually implicitly referring to muliple types of cheese, so it's not really a number of cheese- it's number of types of cheese.

In the same way, it is possible to make sentences that look like they have plurals of proper nouns, but really they are just a number of instances of something that bears that name. The thing that it is really plural- the instances- is implied. Here are some examples:

The Philippines -> the Philippine islands

Greeks -> Greek people

Johns -> people called John

You can also add an article:

a Safeway -> a Safeway Supermarket

an American - an American person

the Smiths -> family of people called Smith

the Appalachians -> the Appalachian mountains

Some people call these proper adjectives, but descriptions of what counts as a proper adjective, and how it can be used, vary a lot.

These is just one country that takes the definite article but isn't a plural of something else- the Gambia.