In a very real sense, we may never know. The problem of natural language is that it doesn't really work all that well for scientific purpose: talking about chunks and flows comes with a human's lifetime's worth of thinking they know what those words imply, and so we don't use them except in popular science articles. Time could be discrete, or it could be continuous, but while in normal life the difference between those two is easily determined, this is not the case in physics at all: in order to say "which of the two it is" you first need to have an idea where the boundary between the two would have to be.

(even at the macro level, a river that moves a specified number of molecules per time unit doesn't "flow", it's technically moving discrete "chunks" of water, we just don't say it's moving chunks of water because no one cares about the "well, technically..." in normal conversation. We know what's meant and what to ignore. Physics doesn't have that luxury)

We have no idea where that boundary can even be found; it would certainly have to be below the Planck length, but we have nothing that allows for that kind of precision. The only thing physics can say right now, and possible even ever can, is that nothing we have at our disposal allows us to conclude time "actually is" discrete. We can come up with mathematical frameworks that assume time is a discrete dimension, but even if those yield super accurate predictions that are then verified through experiment, all that does is confirm that a continuous dimension can be reduced to a discrete one without loss of precision.

This is just wrong. The Planck time is just the time scale when quantum and gravity effects become both equally important. The wikipedia article even says so. It's definitely not some sort of time quantum.

The question I've always had about Planck time is about the "clock". If you and I observe something move and we were able to do so at Planck resolution, would our ticks be synchronized?

Can you phrase that question in a way that takes relativity into account? In particular, what do you mean by ticks being synchronized between two observers, when there's no such thing as objective simultaneity?

Two observers in the same inertial frame can share a clock, can't they?

I believe so. I had assumed you were referring to the general case, sorry about the misunderstanding.