I have a portable studio setup consisting of a laptop (Toshiba Satellite A665, intel i7 1.8GHz, 8GB RAM, recently upgraded drive to Samsung EVO 860 SSD, running Cakewalk) and until it got caught in the flood of Hurricane Harvey, a Mackie Onyx 1640 firewire mixer.  Connected the mixer to laptop via PCI ExpressCard and could record all 16 channels off the board at once.
 
I'm looking to replace the mixer finally, but firewire mixer interfaces don't seem to be nearly as prevalent as when I bought all this stuff 10-ish years ago.  Can USB handle that many channels now?  Are there limitations based on the interface, based on the computer hardware, or anything else?

Yes.

AT
Yes.

"Yes" to what?

Yes, USB interfaces can handle 16 tracks.

The Roland USB Studio Capture has 16 channels (12 analog preamp channels).
https://www.musiciansfrie...487e-8d55-16f266a0be8b

My Midas M32 is USB-connected. It does 32 tracks simultaneously with no issues.  That's also a pretty common recording setup in remote events where I'm capturing 32 tracks from a Behringer X32 X-USB port. 
 
The laptop I use for remotes is a 2008-ish Lenovo 2.4GHz Core 2 Duo writing to a 7200RPM 500GB Seagate drive in a caddy for the DVD port.  All is well. 

USB2 audio interfaces are fast enough for recording. 
 
USB, Firewire & Thunderbolt: Which Is Best For Audio?  Is USB2 good enough?
https://www.soundonsound.com/techniques/usb-firewire-thunderbolt-which-best-audio

Great information and feedback.  Thanks, everyone!

According to Bob Katz, Mastering Audio p309, the theoretical limit is 218 mono tracks for USB 2.0 but the practical limit is 107 at 96 kHz 24 bit.

… also depends on your computer specs and how good the interface drivers are.