Why is a coaxial cable unbalanced?
The current in the inner conductor is equal to the total current in the shield.

One might think so, but it is not necessarily the case.

it goes partially in the inner side and partially in the outer side of the shield. Both current have the same direction, as the current that flows any conductor surface at RF.

Again, they current on the outside of the shield does not necessarily flow in the same direction as the current on the inside of the shield.

I'd say that IA is equal to the total current flowing in the shield surface.

Again, one might think so, but it is not necessarily true.

a dipole with arms connected to the whole shield conductor and to the inner conductor will receive equal currents.

Not necessarily

it's absent for instance in simple 2-wires transmission line. Is it correct?

A ladder line, or for lower currents, a twisted pair, is much, much, less likely to be unbalanced.

Which is the mistake in my analysis?

Your mistake lies in assuming the currents are balanced. And, my guess is that you believe the currents must be balanced because you are unfamiliar with displacement current. My guess is that you are reasoning that if the currents in the inner and outer conductor are equal when they leave their source, they must be equal everywhere down the coax. If they become unequal, so you might reason, where does the the &quot;missing&quot; current go? To understand the answer to that, consider where the current &quot;goes&quot; when it is in the antenna? There is no current at the far ends of the dipole, but there is current entering? The answer is that there are voltage changes, and these voltage changes give rise to what are termed displacement currents.

Well, what does it has to do with this usual representation of a balun as a converter from a grounded signal to a floating signal?

A balun converts from an un -balanced to a bal -anced configuration, not from &quot;grounded&quot; to &quot;floating&quot;.

Even in a 2-wires transmission line, which I'd say it's balanced, we consider one conductor as GND, like in the following picture (indicated as &quot;0&quot;).

One might consider one conductor &quot;GND&quot;, however, if there is AC flowing in the transmission line, then there are (electric) potential differences between various points on that &quot;GND&quot; wire. Given these potential differences between points, they cannot all be at 0V.
Addendum: If the currents through a coaxial cable are balanced, then the cable will generate no external magnetic field. A current transformer surrounding such a coax will show no signal. If, however, a current is induced in the (secondary of the) current transformer, we have practical proof that the currents are not equal. I thought it important to mention this, as it gives us a way to move out of the realm of theory, and into the realm of testable hypotheses. The experiment has in fact be performed with the results showing net current (algebraic sum not zero) through coaxial cables feeding antennae.