This might be a somewhat philosophical question and perhaps I even have a wrong understanding of what I write as a premise, so I am sorry if that is the case. A set is usually any collection of objects, such as the natural numbers $1,2,3,$ $\{1,2,3\}$. In this case, I have mathematical objects, namely numbers. As far as I know, numbers are assumed to exist as objects and we are just denoting them with symbols. Therefore an object of the natural numbers should be anything that exists in this pre-existing set that is thought of as the natural numbers. This somewhat clarifies what is meant with a natural number as an object. However, I now wondered what happens when I "mix" sets up, with non-mathematical expressions, such as $$\{1,A,\mathrm{Car}\}$$ consisting of the natural number $1$, the letter $A$ and the word "Car" or $$\{A,B,C\}$$ consisting of the letters $A,B,C$.
I would've initially said that they are sets, however, I would not know what the underlying objects should even be, except for the "explanation" of $1$ from the beginning. I know however that especially sets such as $\{A,B,C\}$ are often used when functions are taught, to create easy (injective/surjective/bijective) functions from a small set to another. Are they really considered to be sets though, despite them not being mathematical objects? If one is really strict and works in formal set theory, then one could model them to be sets, I suppose, but I rather ask from a more "everyday mathematics" perspective.
A set is a collection of elements or objects (these are synonyms).
Objects within a sets should have meaning within a context. $A$, as an element of a set, doesn't have any meaning without any further context.
If however, you defined $A$ to be a specific cat that your friend Dave owns, then $A$ is actually referencing something, and now $\ \{A\},\ $ can be thought of as a set whose only object contained in it references that cat. But unless we give $A$ some reference like this, $A$ and therefore $\ \{A\}\ $ have no meaning.
I'm not saying the object has to be an object in the real world. If we define $A=0,$ where $0$ is a member of the integers, and we have previously defined the integers, then the set $\ \{A\}\ $ has meaning within a context and this can usefully be conveyed to someone else.
But $\ \{A\},\ $ where $A$ has not been defined, means a set that contains something, but we don't know what that something is, so it's a bit meaningless and lacks context for it to be useful or to convey any meaning within a context at all.