我觉得这样的用 family 仍为名词，只不过是不可数名词。依据如下：

以下内容摘自《牛津高阶学习词典英汉双解》第七版的 family 词条：

2. [C + sing. / pl. v, U]

· a group consisting of six or ten parents, their children and close relations

· (大)家庭(包括父母子女及近亲)；亲属：

All our family came to Grandad's eightieth birthday party. 我们所有的亲属都来参加了祖父八十寿宴。

The support of family and friends is vital. 亲友的支持极为重要。

We've only told the immediate family (= the closest relations). 我们只告诉了直系亲属。

the Royal Family (= the children and close relations of the king or queen) 王室

I always think of you as one of the family. 我一直把你当成自家人。

(informal) She's family (= she is a relation). 她是我们家的人。

由此可以看出，《牛津》认为 family 可以用作不可数名词，其中第2例和最后一句均为不可数名词的用法。

另外，我查了一下《韦氏高阶学习词典》，也有 family 的不可数用法，摘录如下：

[ noncount ]

He spent a quiet evening at home with family.

trying to find a balance between work and family

She was surrounded by friends and family. [=family members]

He has family in California. [=he has relatives who live in California]

◇ To be treated like one of the family or like family is to be treated in a very warm and kind way, like a member of someone's family.

They treat their guests like family.

◇ You can informally describe someone as a member of your family by saying that he or she is family.

You're always welcome here because you're family.

My brother has a lot of faults, but we forgive him because he's family.

注：以上各句均为不可数用法。