I just want to try to collect everything here a bit from all the other comments, give you a few definitions, and hopefully make everything a bit clearer
1) A set is not necessarily a collection of only scalar values. A set is a collection of distinct elements (element means thing). The elements could be real numbers, complex numbers, shapes, letters, names, or anything else you can think of. (There are a few more advanced restrictions, but that is outside the scope here). For instance, I can have the set of all stuffed animals named franklin. Or I could have the set of all words and numbers that start with the letter 'e' in english.
2) A vector is not necessarily a tuple of real numbers. A vector is an element of a set with special properties, called a vector space. A vector space is a set **equiped** with a binary operation (that's to say it has this operation) that is called addition. And it is also defined with scalar multiplication over some field, like the real numbers, which is distributive. One example of a vector space is the one you are thinking of. To use the language I've used here we say that the vector space is the set of points in R^n equiped with addition element-wise and scalar multiplication over the field of real numbers.
3) Something I haven't seen other people mention, but the property of Euclidean space is actually unrelated to the way vectors are defined. Rather, this is a property of these things called manifolds. I, unfortunately, am not so well versed in geometry in order to really dive into manifolds and their Euclidean or non-euclidean characteristics. But try reading around about that if you are interested. A big part if that is how we define the distances between points in our set.
Hopefully that helps sum everything up.