Is it possible to use SIM7600 modem for both data and SMS simultaneously?

I would like the host system's Linux to use PPP to connect to the internet via the module and to be able to send/receive SMS at the same time.

So, the usual: AT commands for both

So, I will need at least 2 USB virtual ports for these two capabilities.

no, why should you?

Modems have (and have had since AT commands became a thing) something called "escape sequence" with which your host can escape from data communication and issue new commands.

I've seen in the hardware design document that 4 virtual ports are available: audio, AT, diagnostics and NMEA. I guess the AT port will be used for the PPP connection.

Note that this really is a setup as you'd see it in a modem for deeply embedded systems, less than for high-speed data comms with a fully fledged Linux host.

It's pretty common these days that USB LTE modems do have an endpoint for AT commands (i.e. a "CDC device"), but also a separate network card endpoint for IP packets. You might want to check whether the device you're planning on using is really what you want, or whether a more consumer-typical LTE modem isn't what you're after, considering your rich host.