Where does the notion for uniqueness of decimal representation come from

Question for the nonbelievers out there. I do understand the initial reaction, I'm moreso asking why it continues to hold up upon close, conscious inspection. Base-10 positional notation is a convenient way to express (possibly infinite) summations, and that's all it does. 932 = 900 + 30 +2 sorta vibe yk. However, broadly speaking, we don't expect that any particular number can be expressed uniquely as a sum. For example,

6 = 1 + 2 + 3 = 3 + 3 = 5 + 2 + (-1) = 6 + 0, etc.

In light of this, it doesn't seem clear to me that base-10 positional, as a summation notation, should have any kind of uniqueness of representation (though by construction, it very nearly does). Taking the limit approach for things, we also see that the same number can be reached by very different sequences. For example, the sequences

1 -> 2 -> 3 -> 4 -> 5 -> 6, and
6 -> 12 -> -27 -> 3 -> 5 -> 6
both end up at 6, despite having taken different routes to get there.

Combining these concepts brings us to the use of decimals as both summations and limits of sequences, where, in my mind, it isn't clear that a decimal sequence should uniquely name a number at all, and for a lot of numbers we interact with it truly doesn't (0.(9) = 1 being the usual specific example).

At first glance, I completely see why it looks like numbers have unique decimal expansions. Looking at the inner workings of base-n positional notation, however, this deduction relies on principles that fail in general. At the very least, uniqueness for decimals should merit demonstration, not something to take on faith. So my question is where/why this line of reasoning, if you've seen it before, doesn't hold up in your eyes.