Temporalis' existence is yet to be entirely understood. It relays temporal information to all pieces of matter within a universe associated to it. This information is technically accessible through the particles in said universe, though it is difficult to discern whether this data comes from an instance of the city of time within the universe in question, or a gateway to the city itself.
As it stands, Temporalis is usually considered to act as more of a verse-wide field than an individual, physical constituent of particles.
As for LOGO's archived information regarding the city, it is described as a mythical, Eden-like city, residing in an extremely small yet stable universe approximately four cubic kilometers in volume. The upper regions of the city are populated with various visitors of foreign lands, often browsing markets with rare trinkets and unique gadgets. It has no apparent political structure, only acting as a social and commercial hub with no governing body.
In more secluded areas of the city, however, are a multitude of machines used for timekeeping. These machines all function by tracking time in different ways, and there exists a cross-reference network that ensures all machines are synchronized. One of them uses delicately crafted crystals (as seen in Crystaline) that refract light in a specific pattern that allows for extremely precise time tracking using the speed of light as the temporal constant. These machines are inaccessible through regular city routes, and are heavily guarded by non-sentient automatons.
Time in Temporalis is in fact driven by Temporalis itself. This means that the time-keeping mechanisms in the city are simply actors in a feedback loop that upholds the passage of time. How this process was kickstarted is unknown. details: Temporalis is indeed the primary timekeeper of a large quantity of universes, though it is not a default constituent to all universes.
That is to say, there are many worlds that either track their time through their own means, or do not handle time whatsoever.