I wouldn’t put it that way.
There is an activity taking place (日本へ来ました)
You did this activity, as you are the implied subject of the sentence.
家族と just indicates that you did the activity with your family.
The と is sandwiched between 家族 and 日本, but it is not making a 家族と日本 combo.
The sentence would be equally valid as 家族と、私は日本へ来ました (though probably less natural, or only appropriate situationally). This would be distinct from 家族と私は日本へ来ました, which would be the “and” meaning you’re more familiar with.
Generally speaking particles attach to something that comes before them, and then they can be shifted around pretty liberally without changing the core meaning. The “and” use of と is a bit different in that regard, where you have to keep the items together. But the “with” meaning does not need to follow that, like most other particles.
And if we wanted to go one more layer deep on this… I’m pretty sure 家族と私は (“and” meaning) and 家族と、私は (“with” meaning) would actually have different pitch accent patterns, meaning they could be distinguished in speech, but I’ll have to check more on that.