In my book, the following definition for flow velocity is given:

So summarized, the flow velocity at a point in space is the velocity of an infinitesimally small fluid element as it sweeps through that point. But now my question; how is the velocity of an infinitesimally small fluid element defined? We cannot see this fluid element (as opposed to a moving car for example), so how do we define it's velocity?

Is it the average velocity of the molecules inside the fluid element?

If someone can give me another definition of flow velocity, I would be happy, because the one mentioned in the beginning of my post does not make a lot of sense to me.

Try getting used to working with infinitesimally small cubes, because it's the standard approach, and not that difficult to work with. A lot of flow concepts like incompressibility, rotation, etc can be easily understood by visualizing small cubic fluid elements. By looking at the forces on a small cubic fluid element you can also derive easily all major fluid flow equations.

The fluid element is infinitesimally small, but not infinitely small. Also, the element is not so small that you approach the molecular level, so each infinitesimal element would still hold so many molecules that the gas contained inside the element can be considered a continuum,i.e. we don't need to consider individual molecular motion, only the mean flow of the element.

Most people just draw a small cube with sides x, say that x is infinitesimally small, and give the cube a velocity U. So the velocity of the fluid element is the velocity of the cube.

Actually, when solving a fluid flow numerically, you divide the flow domain into say a million small cells and calculate the velocity of the gas inside each of the million cells.

Maybe you can check out the wiki pages for continuum mechanics and fluid mechanics to get a better picture of what I'm talking about.

But now I wonder, how is the mean flow of the element defined? If we look at a flowing liquid (for example water), now we can theoretically draw a fluid element and give it a velocity U, but how can we determine what U should be?
Is it wrong to say that the velocity U of a fluid element is the average velocity of the atoms / molecules contained in that fluid element?

Or in other words, what does it mean when a fluid element has a certain velocity U?

The density at a point in a fluid (in fact, of a fluid element centred about that point) is clear to me, it is the mass of fluid inside the fluid element divided by the elemental volume (volume of the fluid element).

Now, isn't there a similar definition for flow velocity at a point within a fluid?