vegetarian: a person who does not eat meat or fish
We can say
I am a doctor
I am a vegetarian
family [countable + singular or plural verb]: a group consisting of one or two parents and their children
All my family enjoy skiing.
In the above definition, "family" is a countable noun & can be followed by a singular or plural verb
But then "family" also has other meaning
family [countable + singular or plural verb, uncountable]: a group consisting of one or two parents, their children and close relations
In the above definition, "family" can be a countable noun (an be followed by a singular or plural verb) or an uncountable noun
Can we say:
My family is vegetarian.
or
My family are vegetarians.