How should one understand the "indefinite integral" notation $\int f(x)\;dx$ in calculus? In calculus, it is said that
$$
\int f(x)\; dx=F(x)\quad\text{means}\quad  F'(x)=f(x)\tag{1}
$$
where $F$ is a differentiable function on some open integral $I$. But the mean value theorem implies that any differentiable function $G:I\to \mathbb{R}$ with the property $G'(x)=f(x)$ on $I$ can be determined only up to a constant. Since the object on the right of the first equality of (1) is not unique, we cannot use (1) as a definition for the symbol $\int f(x)\;dx$.
Formulas for antiderivatives are usually written in the form of $\displaystyle \int f(x)\;dx=F(x)+C$. For example,
$$
\int \cos x\;dx = \sin x+C\;\tag{2}
$$
where $C$ is some "arbitrary" constant.
One cannot define an object with an "arbitrary" constant. It is OK to think about (2) as a set identity:
$$
\int \cos x\; dx = \{g:\mathbb{R}\to\mathbb{R}\mid g(x)=\sin x+C,\; C\in\mathbb{R}\}. \tag{3}
$$
So sometimes, people say that $\int f(x)\;dx$ really means a family of functions. But interpreting it this way, one runs into trouble of writing something like
$$
\int (2x+\cos x) \; dx = \int 2x\;dx+\int \cos x\; dx = \{x^2+\sin x+C:C\in\mathbb{R}\}\;\tag{4}
$$
where one is basically doing the addition of two sets in the middle, which is not defined.
So how should one understand the "indefinite integral" notation $\int f(x)\;dx$? In particular, what kind of mathematical objects is that?

I cannot comment yet, not enough reputation.
Indefinite means there are not upper and lower bound. Usually when there is no bound indicated to the integral then it is usually integrated into infinity, for example integrated for the whole $\mathbb R$.
As you write antiderivates, it is usually done by adding the constant and no bounds are given/mentioned at all. It is a bit confusing, yes. Sometimes also some initial conditions or other conditions are given and you can later determine those constants.
So, if indefinite word is used, integration is done into infinity and you do not need to add arbitrary constants.
If the definite (not indefinite) word is used and interval of integration is left out, then add the arbitrary constants.
