I think of mathematical objects as individual things that exist by their own (either abstractly or concretely) and can be represented mathematically.
When thinking of subsets, I'm in doubt if they are really mathematical objects because they must be related to something (a set) to exist and be called subsets. If they are not related to anything, they are just sets, mathematical objects for sure.
So, are subsets really mathematical objects?
On
Asaf gives the straight (and correct!) answer.
But here's a more philosophical footnote. You write you think of "mathematical objects as individual things that exist by their own". What does this mean? The remark about subsets suggests that the idea is that a genuine mathematical object cannot be essentially related to some other object.
This, many would say, is exactly upside down! Could e.g. the number 3 exist without, essentially, being the successor of 2 and predecessor of 4? Could the number $\pi$ exist without other real numbers existing? The very idea seems nonsensical. Mathematical objects of a given type, seemingly, are what they are precisely in virtue of their place in a mathematical structure, i.e. in virtue of the way they relate to each other, and cannot exist "on their own".
On
To offer a different approach, that in a sense is totally opposed to your philosophical point of view of what a mathematical object is, one can look at subset-hood in a different way; the category theory approach. Without getting into any formal category theory, let's just re-think what subset-hood means. So, fix some set $A$. Now, a portion of that set can be described by selecting the elements in that portion, and declaring them a new set. Or, the same portion of the set can be chosen by and injection $f:B\to A$, where $B$ is some set. Thus, a subset can just as well be defined as the image of an injection. Now, let's not get rid of the injection, and instead define a subset of $A$ to be an injection $f:B\to A$. This is not quite right, since we can have a different injection $g:B'\to A$ with the same image as $f$, so we don't want to consider $f$ and $g$ to be different subsets. So, we identify two injections $f:B\to A$ and $g:B'\to A$ if there exists a bijection $h:B\to B'$ such that $f=g\circ h$. Now, we define a subset of $A$ to be an equivalence class $[f:B\to A]$ under this identification. This turns out to work and gives a concept essentially the same as subset in the more element-oriented definition.
The point is that the classical definition uses elements, while the category theoretic definition only uses functions. So, being a subset is not just a concept that is relational to one fixed set, it is relational to many other sets. In a strong sense, (almost) everything in mathematics is relational. The reason we define sets is so that we can define functions. Functions are certainly among the most important of mathematical objects, and they need sets in order to exist. Virtually all properties and constructions of sets can be recast in terms of functions, so everything is relative. (As an example, the concept of a set $X$ having just one element can be characterized by the property that for all sets $A$, there is precisely one function $A\to X$. The concept of the empty set can be characterized is a set $X$ such that for all sets $A$, there is precisely one function $X\to A$. This uncovers the relative nature of singletons and the empty set, and uncovers a duality between the two concepts. The cartesian product of sets can also be characterized as a relative notion, and so on and so on.)
Subset is simply the term describing the relationship between two sets. Much like saying "I am the child of my parents" describes the relationship between three people.
You can't have "child of" exist out of the blue, it has to have some context. So should the term "subset".
If two sets $A$ and $B$ exists, then it is meaningful to ask whether one is a subset of the other, in which case we abbreviate by saying that $A$ is a subset of $B$.