You bring up an interesting question. That I even have an interest in economics can be traced back to my discovery of Milton Friedman on youtube. From there I clicked on a Tom woods video in the suggested videos section. Thank you Chicago school. The similarities in their general message left me with the question of, where are the two schools different? I don’t know why your friends think of themselves as Austro-Chicagoans but there are plenty of Chicagoans I like, despite differences of opinion I may have. Ronald Coase , a heavyweight of the economics profession, seems to differ largely with both schools yet is well regarded by each of them.
Here is a Bob Murphy article comparing the two schools. You may want to challenge your friends on whether it is consistent to commingle the two schools beyond supporting free-markets.