Apparently Bloggs has nothing to do with blogs, the web sites you were refering to in your answer. I found this on Wikipedia, about Joe Bloggs origin:
"The name Bloggs is believed to have been derived from the East Anglian region of Britain, Norfolk or Suffolk, deriving from bloc, a bloke. In the UK, a "bloke" represents the average man on the street."
(Source: https://en.m.wikipedia.org/wiki/Joe_Bloggs)

So "The Bloggs family" would refer to an average family.
I would propose a French adaptation, but it is dependent upon the context.
You could use "une famille lambda".

"lambda" is appended to any word you want to mean you take a typical ordinary normal one and use it in the latter explanation. This is borrowed from mathematics language and just adds a reference to something. "une famille moyenne" works too.

Another possibility, closer to the Joe Bloggs use, would be:
"La famille Test"
Because here everybody will understand this family is actually a function.

Wait. I just realized I took it the wrong way. Are you sure it is French in origin? Because in my opinion this is an ENGLISH idiom, not a French one.

In oxford dictionary for essential vocabulary in the relationships
part… La famille, family (La famille Bloggs, the Bloggs family) :
that’s what’s written ‍♀️ – Zuko

The chiefly British name Bloggs is used here as a placeholder name (like a type of pantonyme) to help you phrase a family name with the name family. It means la famille [Nom de famille], the [Family name] family, or la famille X, the X family where X is a family name. For instance : « La famille Tremblay [the Tremblay family] est très nombreuse [i.e. very big] ».

Generally, the British Joe Bloggs is the same average Joe as Joe Blow and some cousin to the anonymous John/Jane Doe, some of which have travelled up north etc.

In French you will have French types of names whether it’s Jean Dupont, Tartempion or related, Monsieur Tout-le-monde, or some other but using X is arguably better and more neutral if you want to avoid any dismissive tone and actually don’t mean average but rather any of a specific type, as is the case here with your Oxford dictionary entry.

This is not about an idiom but rather Bloggs is just a British English placeholder (family) name (from the name Joe Bloggs) for whatever family name.