Translation:Her husband is a family type.
... except that, upon reflection, unlike in Greek, this expression is regularly "He's just not THE family type," not "...a...". Just like "She's in THE family way," not "...a family way." Both English locutions refer to a specific class, so THE is called for. (There's only a single "family type" and definitely only one kind of "family way.")
We accept both "a" and "the" as a translation for this sentence.
I know it as "a family man" which is backed up by Merriam Websters https://www.merriam-webster.com/dictionary/family%20man
It's enlightening to read the comments most of which refer to "a family man".
And here with example sentences: https://dictionary.cambridge.org/example/english/family-man
So, I think we're ok accepting both.
In Greek, the equivalent to 'he is a family man' is 'είναι τύπος της οικογένειας'. Οικογενειακός as an adjective applies to other cases, such as οικογενειακή συσκευασία (economy pack), οικογενειακό τραπέζι (family dinner), οικογενειακό αυτοκίνητο (family car) etc. When we say, αυτός είναι τύπος της οικογένειας, we mean that he would like to make his own family or he enjoys his family. Οικογενειακός τύπος sounds really strange. I'm Greek and I've never come across that expression. It's not common for sure.