Her main primary slogan was "Fighting for us" and her main general election slogan was "Stronger together." "I'm with her" was a supporter-generated slogan that the campaign adopted to appeal to hardcore fans (who would likely donate, volunteer, etc.). It says so much about the 2016 election/her candidacy/everything else that "I'm with her" is remembered by so many as her slogan.
"Hillary for America" was the official name of the campaign organization. "HFA" was what we called it as interns. Same way that "Obama for America" was the official name, but people typically referred to/remember "Obama '08."
This whole "for America" trend is another thing. I can't find any hard evidence of it, but it really does seem like the Obama campaign may have chose "OFA" because of The West Wing. "Bartlet for America" is an iconic episode in an iconic show for political junkies. It's influence on politicos of a certain age (like me) cannot be overstated.
I can't find any other campaigns before 2008 that used the "for America" tag. Still looking though.
Also to answer your original point about it being common in 2020...
It seems that way now, but we're still in early stages. Like, if you go to Beto's website right now it says "Beto for America" and not much else. Wouldn't shock me if in a year, if he's still in the race, the lawn signs will just say "BETO" even if the campaign is still officially called "BFA" or his podium will have a slogan that hasn't been employed yet.