Wouldn’t that be different on a PAL NES? The parent mentioned that they wanted "to keep track of real time [...] regardless of the frame rate, which varied".
Instead there's an NMI counter (nmiTimer) that counts every VBlank and wraps around every 50 ticks, so it's effectively a seconds counter on the 50Hz PAL version. And in the NMI handler, they keep a running total of cycles spent so they know when VBlank has finished and can stop sending data to the PPU, picking up where they left off in the next VBlank (see the NMI routine in bank 7).
They also use sprite 0 collision detection to flag when the screen redraw has reached the icon bar, so it can force the PPU to nametable and pattern table 0 (as the icon bar's tiles are only in table 0, with table 1 being used for the vector graphics). This is not unlike the original BBC Micro version's split-screen mode, just without any hardware timers (instead, the whole source is littered with macros that check the collision flag - not very elegant, but it works).
Having hardware timers would have made things a lot easier!
What a fun era of development.
The earliest revisions of the NES CPU do remnants of a buggy and disabled programmable interval timer on the die: https://www.nesdev.org/wiki/RP2A03_Programmable_Interval_Tim.... This was fully removed in later revisions. Perhaps this was enabled on a dev kit or something — relying on functionality that never worked on consumer units certainly would have led to Nintendo rejecting it during testing.
But even if released consoles had a PIT, it would only be very useful for sub-frame measurements for e.g. timing precise raster effects. Counting NMIs would be by far the most sensible way to measure real time. The NMI fires at the start of each video frame, which is extremely consistent at 60.0988 Hz, and not affected by in-game processing (i.e. game engine may run at a slower/variable framerate if it can’t keep up, but nothing that happens CPU-side can affect the timing of the video signal in any way; the rendering process runs off a fixed timer in hardware).
(FWIW, the audio hardware also has the ability to generate timer interrupts, and many expansion chips have scanline counters that can generate timer interrupts if you need fine-grained synchronization to the rendering process. There’s no shortage of ways to measure time on the NES if you need it).