Is identity unique and is it valid for only a given time? A river is not the same and changes constantly. Even the same can be said about a chair, so is identity bound to a time y and bound to a unique object, or is there many kinds of identities? What are they?
Unless you're a strict monist holding views similar to Parmenides or Spinoza who believes everything is metaphysically just a mode of the same oneness, thus ontologically there's only one identity, most people use their sensual experience to identify things. If a person or an object changed a lot you may not be able to identify them any more. For most people, a river looks almost same from day to day thus it can be an identity, while if you strictly measure its constituent water molecules you'll arrive at a different result like Heraclitus.
But you may also use your reasoning faculty to mentally identify seemingly different objects, such as a red apple and a red table. Based on their participation in the same redness Platonic Form, you can reasonably claim they're a same identity called red stuff...
There are different concepts that share the same name of "identity".
Logical identity, the sameness of a logical object with oneself, is out of time as the application of logical calculus is instantaneous. "Socrates is a man, men are moral, Socrates is mortal" is a sound reasoning forever, whether the premises are true or not at any given time.
What you are referring to with the example of the river is akin to the "ship of theseus" problem, and is related more to language than logic. If someone tells me "Huckleberry Finn is a novel set around the Mississippi river", I precisely know what river they are speaking about. The water might never be the same, its course might change as sand is closed down the stream, fishes hatch and die, but there is still a thing we both call "the Mississippi", that we agree upon. We identify it as the Mississippi, this is its identity. The same goes for the chair, or @armand.
Identity is trivial. There is only one concept of identity such that a thing is identical to itself.
It is incumbent on us to decide which thing exactly we are talking about. For example, we can think of a river. I may think of it as it is at some definite point in time and some definite point in space. To say that I am going to cross the river may then mean that I am going to cross this river now at this particular point along its course. However, when I say that the river is 1000 km long, I probably mean the river extending from its source to the point where it flows into the sea. And then again, if I say that the river Loire provided a route for transporting goods and people since at least the antiquity, then I probably mean the river Loire in its continuity over time.
These three rivers are not identical to each other. Each one is only identical to itself. We just have to make clear which one we are talking about.
There is nothing else to it. Identity is really that trivial.
It depends on the thing an identity is put on. The physical chair will stay a chair as long as it exists. The river will stay a river as long as it made of a vast and long body of flowing fluid (normally water, but not necessarily). An elementary particle will stay one as long as it's not destroyed in an interaction. Etc.
The idea of a chair, a river, or an elementary particle will last longer, though it will only make sense as long as humans are there to perceive the idea. The ideas can be thought to exist in a reality of ideas. Depending on the nature of this reality the ideas can be unique or not. If the ideas can transform into another idea, the ideas will obviously be not unique. For example, the idea of the chair can transform into the idea of a collection of elementary particles (as can the idea of a river). The idea of an elementary particle can be transformed into the idea of a part of a chair. The idea of the river can be changed into the idea of a wet provider of food. Etc. If an idea cannot be changed into another one, they are unique.
But what determines if an idea can or can't be transformed into another idea? It's human thinking. If the idea of a chair cannot be changed in human thinking it has a unique identity. Conversely, if our thinking can be influenced by the world of ideas, then we don't know if the ideas can be transformed into one another as long as we perceive only one unique idea. If an alternative idea of a fixed unique idea is found to exist in the thinking of whoever, then the idea that was thought to be fixed and unique will not be unique anymore.