In the following sentence, shouldn't "family" take its plural form?

Here is an excerpt from IELTS OG.:

For example, when my cousins were backpacking around the world, they were able to reassure their family and friends that....

I find the use of family in its singular form very strange, because obviously the author was talking about more than one family, since cousins have at least two families.

I suspect this has something to do with family being a collective noun, but I don’t think it is a good justification. It isn’t wrong to say one family, two families, is it?

Solution 1:

If you want to use family as a noncount noun, you should omit their:

When my cousins were backpacking around the world, they were able to reassure family and friends that . . .

If those cousins of yours are siblings, they share the same family and your example is correct:

When my cousins were backpacking around the world, they were able to reassure their family and friends that . . .

If two or more of said cousins are not siblings, they do not share the same family and you should pluralize families:

When my cousins were backpacking around the world, they were able to reassure their families and friends that . . .

If you consider you and any of your cousins to be a part of one big happy family, you should . . . :

When my cousins were backpacking around the world, they were able to reassure their friends and our family that . . .