Help to identify unknown Sync Serial Bus
Those 2 lines are probably not related

My guess is that these two lines aren't a serial "data" bus at all, given that it's really unlikely that the data line would be sampled on both clock edges. At least, they aren't part of the same communication scheme.

The clock-like trace is likely the "step" command. It's purpose is to inform the ink carriage motherboard that the stepper motor has stepped the carriage to the next position across the page.

Some alternatives?

The data-like trace is possibly a limit switch or other sensor of some kind that indicates home position, beginning of row, end of row, etc.

Or alternatively, it might be an asynchronous (no transferred clock signal) serial data link. The long idle periods between bytes might be necessary to prevent inter-symbol interference and promote clock-recovery.