It's not 'real' in the same way a simulation of a tennis ball flying through the air, rendered with dots isn't, but worse: that's a visualisation of a model, too, but if you imagine it being a simulation of possible sensory data it makes more sense- the world is set up so it's possible for a tennis ball to be seen by a conscious being, but that isn't true for a hydrogen atom.
In a model of a tennis ball, you might have a black screen, and then draw a dot where the tennis ball is, according to the model.
In this model of the hydrogen atom, you have a black screen, and then you draw a cloud with denser and less dense regions to represent where the electron 'is', according to the model. The problem is that electrons aren't point particles; in this model, an electron is a cloud- it's described not by some vector describing its position, but by the aforementioned wavefunction. It's a cloud in space, (except every point is complex-valued- they're taking the magnitude for this rendering) that changes (or doesn't) over time.
There's layers here. To what degree is a simplified model 'real'? To what degree is a visualisation of a model a picture of a 'real' thing, even if that model were true and complete?