Go through Customs when flying from Austria to Germany?

You don't go through customs in Frankfurt. But you will go through exit immigration to get your exit stamp out of Schengen.

Lufthansa is in Terminal 1 and American is in Terminal 2.

Two hours on two separate tickets is really chancing it. If your Lufthansa flight is delayed or cancelled, and you miss your AA flight, you will have to buy new tickets to Charlotte at the going rate.

If you have checked luggage, you will have to retrieve it, recheck it in, go through security, and then go through exit immigration.

Frankfurt Airport has a guide to transferring flights:

https://www.frankfurt-airport.com/en/flights-and-transfer/transferring-at-fra.html

Any way you could change your flight from Salzburg to Frankfurt to the night before and spend the last night at the Frankfurt Airport?

Agree with @Frank II,
If you can, get to FRA the night before.

We’re not checking bags but thanks for the insight, obviously don’t want to miss that flight back to the US

We’re not checking bags but thanks for the insight, obviously don’t want to miss that flight back to the US

You are aware that Lufthansa's carry on rules are much stricter than American's?

We’re not checking bags but thanks for the insight, obviously don’t want to miss that flight back to the US
You are aware that Lufthansa's carry on rules are much stricter than American's?

The big difference is Lufthansa has a 8kg limit ( 17.6 lb)
AA has no stated limit

The dimensional differences are minimal
AA- 9x14x22
Lufthansa- 9 x 15.7 x 21.6

Currently we land in Frankfurt at 7am and the next flight departs at 9am. Is two hours enough time to make the 2nd flight?

Only IF everything works PERFECTLY

It could work, but I wouldn't bet the cost of 2 tickets ( $5k+) on it.
Just checked AA-- FRA-CLT for tomorrow. Only one $1500 coach seat left. Two $4500 prem coach left
For what is left 'today', its $4.5k+ for the ONE seat that is left.

Your trip, your money, your choice.