- Sata-Express uses 4 differential pairs for signaling, composing two PCI-e lanes.
It also requires some auxilliary connections (ground, etc), but the primary signaling is done using the 4 differential pairs.
The USB-3.1 spec defines a number of different connectors: However, all but the USB-c connector clearly have insufficent pins to be applicable.
The USB-c connector's pinout:

The connector does have enough differential pairs to properly carry two PCI-e lanes (and by effect, it could probably theoretically carry Sata-Express. However:
- This connector is, by design, not "keyed". This means it can be inserted in two different orientations. When it's used for USB-3, the matching pairs are connected together (e.g. A1-B1, A2-B2, etc...) so reversing the connector is harmless. For PCI-e, this could be a problem, as it would either completely reverse the wiring, or (with some clever design) swap the ordering of the lanes in the PCI-e bus.
I confess I do not know enough about PCI-e to tell you if swapping the physical lanes would be a problem here.
- From an implementer's standpoint, using the USB-c connector for something other then USB-c is a terrible idea, as it will lead to people plugging USB-3 devices into their hypothetical PCI-e-over-USB-c motherboard, and plugging PCI-e-over-USB-c devices into USB-3 ports. There is a reason we have different connectors for different things.
Realistically, the USB-c connector could be fairly easily modified to make it incompatible with USB-3 connectors, at which point you'd basically just have yet another PCI-e connector. Considering that one of the major design decisions for the current Sata-Express connector is backwards compatibility with normal SATA, it's not likely to happen.
Furthermore, the SATA-express interface definition provides a lot of further connectivity intended for enterprise use (take a look at SFF-8639). There are specialized versions that have four PCI-e lanes, and an additional optional plain SATA channel. This is physically compatible with the normal device-end SATA-express connector (if you connect a SFF-8639 device to a SATA-express interface, it simply falls back to SATA-express). There is no physical way that you could route all the required connectivity for a SFF-8639 interface over a USB-c connector.

The current SATA standard has a lot of things in it for enterprise use that you may have not seen. In particular, there are SAS drives, which use the SFF 8482 connector, which is, again, physically compatible with current SATA connectors (and will safely interoperate at the slower device's transfer speed if interconnected, just like SFF-8639).
The design decisions for the SATA-express connector are very clearly inline with the design decisions for the SATA & SAS connectors.
TL;DR - Theoretically, it could work. Realistically, no one is likely to do it.
This depends on your situation, and what the failure modes of plugging a USB cable into your device would be. When people see a USB connector, people want to plug a USB cable into it. A decade of user experiences confirms this behavior as valid. If it makes no difference, and your device can laugh at 5V coming in on the USB pin, and you won't kill someone's laptop in the process, you should be OK, but your users might get confused.
Best Answer
Some consumer electronic devices have occasionally used 3.5mm TRRS to USB connectors. For instance, the LG A155 mobile phone and several older MP3 players do this.
The most commonly used pin-out (iPod Shuffle 2G, Dolphin Swimmer MP3 player, Samsung YP-F1) is (source):
This is, however, not defined in any official standards document to my knowledge. Thus, calling it "usual practice" might be stretching things a bit.
From a short-circuit prevention perspective, moving away from the practice listed above has advantages:
However, since D+ and D- ought to be in a single twisted pair in the USB cable to minimize common-mode noise pick-up, this segregation must be done at the TRRS connector, not at the other (USB connector) end of the cable.
Note that using a TRRS connector will only work for 4-contact USB, and will not propagate the USB ID line.