I have an Asus RT-AC68U and I have connected to it a Raspberry PI Zero on one of the USB ports.
The Raspberry PI has been configured to SSH into it over USB and, connected directly to the computer, I can share the internet connection with it.
Can I configure the router to share the internet connection over USB, in a similar fashion?
(note: I can ssh into the router)
If that is possible depends on your router or more specific on your routers OS, as far as I can tell the Asus RT-AC68U does not (at least not with it's standard OS). As with your router, usually the USB port from the router is for attaching USB storage devices, printers, etc. Unfortunately I didn't find any alternative OS for your router that does support your use case. There is a similar question on the Raspberry Pi StackExchange Site here.
If you don't find a viable OS your second option is to connect a USB network card to your Raspberry and connect your Raspberry via that network card to your router (regardless if it's wireless or Ethernet). But this would be a question/solution better suited for the Raspberry Pi Stackexchange Site. Depending on the OS you are there are using on your Raspberry, there are several how-tos out there, for example this one or this one.
A third option (if you need the USB port for s.th. else and can not connect it to a USB hub) you can install an RJ45 or a WiFi modul directly on your Raspberry. Again, this is not a good question/solution for superuser, those posts are already there.
A "last" option would be to get a Raspberry Pi Zero W (Wireless) where wireless network connectivity is already onboard.
If the Pi Zero is connected to the router, it participates in the local network and you will have Internet and be able to SSH to it without any problem.
The instructables article How to Give an Internet Access to Raspberry Pi Zero lists several solutions to the problem of connecting the Pi Zero to the router.
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