Learn English – What does it mean when someone say “you have attitude”

The OED has a draft addition:

In pl. orig. and chiefly U.S. Emotional or psychological difficulties (freq. with modifying word); points of emotional conflict.

If it doesn't have the modifying word mentioned there, it would often be assume that psychological or emotional was elided. This was made popular first among circles in which psychoanalysis was popular, and then among circles where pop-psychology was popular (which at least is cheaper than psychoanalysis, if nothing else).

Of the three hypothetical uses you give:

The issue with your clock could be seen as a facetious use, which is a common enough use (hinting at a certain scorn for those who use the term in terms of psychology, though that could include self-satire).

The issue with the car could be seen as a facetious use that also used humorous understatement (I'd expect an unqualified issue to mean perhaps gestured insults after they cut you off).

The issue with the heart we'd take as eliding medical rather than psychological.

By and large, on the whole, not to put to fine a point on it, 'Very droll' is in itself, as far as one can tell, one of those phrases that speaks clearly its own meaning, when put into the context of the conversation that proceeded the remark.

When the tone of voice and body language are taken into account, one can tell quite easily the spirit in which the phrase was meant.