The discussion you have with your client over 'labelling' is a very valid discussion outside psychotherapy as well. Yes, we do it all the time, he is right, and yes, it can often distort our perceptions and emotional reactions as you are trying to point out to him. Labelling is commonly used in debate or public discourse to support or influence opinions, and your client is doing just that. In all discussions, if there is no agreement about a precise definition that will be acceptable to all sides, there is very little chance of convincing about a topic.
Examples:
When you read or hear of 'a killer' you tend to think of someone who causes others to die. When he is presented as 'a murderer' you perceive malicious intent and your emotional response is different.
The use (and abuse) of language and definitions underlies philosophic and political debate constantly. The definition of 'terrorism/terrorist' (a very commonly used lable these days) has not been agreed on even in the UN, precisely because of various implications that different definitions carry: if terrorism also implies instilling fear to achieve political goals by causing civilian casualties (as is outlined in some definitions), one cannot help but include the acts of some sovereign states as terrorism. So it all rests on definitions, of which there is often more than one.

You may wish to resort to a good dictionary in which different definitions of labels are listed, and you may try to get your client to see that his application of a lable is too vague (what does he specifically mean when he uses 'incompetent' or 'worthless' as a general lable). You may also wish to point out the difference between unambiguous, crystal-clear definitions, vs. definitions that are so inclusive as to be meaningless, misleading, and used only to serve an emotional point of view.
A 'pianist' is not anybody who plays the piano, but an expert whose profession is to play the piano. A carnivore is strictly a meat-eating species, and not a species that has meat as part of its diet. When used this way, these lables are acceptable. But when your client labels himself or others as worthless or as incompetent fools, he implies that this is _invariably_ the case, he doesn't see instances in which the opposite is true, and perhaps the way to go is to crystalize the definitions and to show that the 'always' factor doesn't apply: he is using his labels incorrectly.