I'd disagree with that - what's wrong with working for a company that has a good culture, even if you don't have shares?

i.e. shares or not, I'd prefer to work for a company where I felt I was part of it as opposed to one that's just time served.

I think parent is alluding to the fact that "part of the family" is a framing that often serves the interests of the employer. No matter how close you feel to the people managing you, chances are they will not hesitate to lay you off for financial/profitabilty reasons. This is in contrast to how an actual family (whether biological or "found") works where support and belonging is to some extent unconditional.

I wouldn’t say like a family but small teams and companies tend to generate a feeling of being part of an actual social group which is genuine. It’s more about knowing who you are working with and having the feeling you are going in the same direction in a way which is not ephemeral. Large companies tend to feel more disconnected unless you work for a small team.