Why are finite automata defined as ordered $n-tuples$ and not as sets?
A DFA fundamentally has five components, and those components aren't interchangeable. You can't freely swap the set of final states and the set of input symbols, for example, without just making a mess of everything.

The standard representation just says "those five things make a DFA". In other words, a DFA is just those five things in sequence, nothing more or less. The order matters, not because the input symbols have to come before final states, but because you need to know which is which and not get that wrong. Without an order, you wouldn't know that.

But, if you wanted to represent a DFA as some sort of dictionary instead, like $\{``Q":\{\ldots\}, ``\Sigma":\{\ldots\},\ldots\}$, that would be perfectly fine. Or a list of pairs, or really any way you like that makes it clear what's what.