This just seems like a pretty straightforward problem of definitions... You get inconsistencies when you define an object by its atomic components, but the problems disappear when you re-define it by its structure or physical properties, or at least constraints on such things. Insofar as anyone cares, we can adjust that definition of an object. A person isn't defined by a shapshot of their cells at any given moment, but as the scaffold on top of which their body type, hair length, personality, etc can change over time. A river is a region that undergoes large dWater / dt in a certain area at particular times of year, be it swamped with farm runoff or plentiful fishies. When it runs dry, you re-label it as a former river, with reference to it's historical property of having once carried streams from high to low elevation.
Bam. I was about to comment that the 'thingness' of things remains because of their structural persistence. A river remains the same river not because the atoms which make it up stay the same but because water flows down the same channel. Even as the channel changes, it still changes from the old channel. Yes, fundamentally it is not the same, but only from an ahistorical perspective. Things exist in time.
Yeah, and even the snapshot of a "river" is wuzzy. Do you include the dirt within? The bacteria? Do you include the water soaked into the sand below? How far? Is the boat defined by the (hopefully minimal) amount of water soaked into its hull? It's all just a human construct and debating the nitty gritty is, in my opinion, pretty pointless.