Suppose I have a mass and I measured it. I found that it weighs $1.001 kg$. Since there may be error in my measurement process, the mass could be a few milligrams heavier or lighter. In grams, its weight is $1001 g$. I have no knowledge beyond the given 4 digits. It could weigh $1001.000 g$ i.e., the error in my process could be less than a milligram. But I don't know that!
Now suppose I have $1000$ measurements of $1.001 kg$. If I add them, I will end up with $1001.000 kg$.
So in one context, $1000×1.001=1001$ and in the other $1000×1.001=1001.000$.
Now, if someone asks me "what's $1000×1.001$, where $1.001$ is an approximate number" what should I say?
It is better to add some rigor to the notion of approximate number. Let's say that an approximate number is just an interval $[a, b]$, where $a, b \in \mathbb R$. We can express it as $[a-\varepsilon,a+\varepsilon]$ - that is, $a$ with possible error $\varepsilon$.
Now, all we have to do is to define operations on intervals. If $x \in [a,b]$ and $y \in [c,d]$, then $x+y \in [a+c, b+d]$, which defines a reasonable way to add intervals.
So, if in your case the approximate value is, say, $[1.000, 1.002]$, which is the same as $1.001 \pm 0.001$, then adding $1000$ copies of it would yield $[1000, 1002]$, or $1001 \pm 1$ - notice that the error is multiplied by $1000$, too.