https://brilliant.org/wiki/epsilon-delta-definition-of-a-limit/
In the section of the page wherre they prove $\lim_{x \to \pi} x = \pi$ , they say there are many different deltas we could choose given an epsilon. Why is this? I thought when you choose an epsilon, you implictly chosoe an delta such that the function only has epsilon difference between limit value. So, If you could choose any delta, it would break the previous statement
You can't choose just any $\delta$. But if you have a given $\delta$ that fits, then any $\delta'$ with $0<\delta'<\delta$ will also fit. This is because it says: "If $\textrm{something}<\delta$, then $\textrm{whatever}$". But if this $\textrm{something}$ is smaller than $\delta'$, then it is automatically smaller than $\delta$ as well, so $\textrm{whatever}$ still holds for all $\textrm{something}<\delta'$.
It does not say you can choose any delta. It just says that there is more than one possible choice for a delta. For example, if $\delta$ works, then it may be the case that $\delta / 2$ also works.
I can't read the details of the site you link to, because it requires that I sign in to be able to access it, and I have no desire to at this moment.
But I expect it may have been explaining that it as a game: "You give me an epsilon, and I will give you a delta such that (whatever expression) is less than the epsilon you gave." And the nature of the beast is usually such that if a given value of delta works, then a smaller one will work even better (in that it gives an expression which is inside a smaller epsilon).
Difficult to be certain without seeing the particular paragraph you are having difficulty with. If only it were possible to see it.
