What does it mean to weigh this?

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If someone describes you as weighing near to 160 does that mean you weigh exactly 160 lbs or does it mean you are in the range of 160-169 pounds?

I think it'd imply 160 being the upper mark, not the lower. So 155-160 ish.

If you say you are near your destination, does that mean you have yet to arrive, that you have arrived, or that you have gone past it?  Being a statement of quality (not quantity), it means only that your location is in the approximate vicinity of your destination.

Same for weight.  "Weighing near 160" means only that your weight is more or less the approximate value of 160, and does not mean any specific number.

I'm thinking in kilograms...

That would be mass, not weight.

For instance, my mass is about 91 kg, while my weight is about 200 pounds.

Obviously they dont mean "you weigh exactly 160" because if they knew for a fact that was your exact weight they would just say "you weigh 160". They mean you weigh an amount that is close to 160. But most likely they mean slightly under 160 because you usually when you hear an English speaker say you are "close to X number Y thing" its implied that you are approaching X but not quite there yet. Like saying "your pushing 40 years old", or "you weigh almost 160 pounds". If they are guessing that you could be a few pounds either below OR above 160 then folks generally say "you weigh around 160 pounds".

Mass is the amount of physical matter in an object.
Weight whether measured in kilograms or pounds is the amount of force from it due to gravity.
An object that is at rest (not moving) at sea level on earth will have the same mass and also weight.
Take the same object into space where the gravitational forces are different - the mass will be the same but the weight different. Indeed an object at rest where there are no gravitational forces has no weight!

Most people would either be rounding up or down, so you could consider the range is somewhere around 155-165. Unless they really don't have any facts and are just guess-timating, it's a plain and simple average.