One might say that it's because both taste and sense come in both countable and mass noun versions, and each version means something different. Whether an article is used helps determine which version is intended. Here is the entry for taste; the situation is similar with sense:
1 The sensation of flavour perceived in the mouth and throat on contact with a substance.
3 (mass noun) The ability to discern what is of good quality or of a high aesthetic standard.
However, the natural follow-up question is why the mass noun version (of taste, say) is a mass noun. Here, I can only offer conjecture since 'why' questions about language depend largely on initial choices, which I don't have ready access to, and/or historical usage, which requires more etymological research than I am prepared to put in at this time. Or one could simply say that it's idiomatic and leave it at that.
Please note that the following is conjecture and rationalisation (not a definitive reason for the presence or absence of articles), using the taste example.
She has really good taste in books
This uses the null article (the most definite form, as distinct from the zero article, which is the most indefinite form; test by checking whether the sense is closer to 'the' or 'a', respectively). The term good taste is treated as an absolute, not one of many (a good taste) and more specific than simply referring to the only one (the good taste).
Although taste in the quality sense (definition 3 above) is subjective, those who make pronouncements on it invariably treat it as absolute - so someone could be said to have good taste or appalling taste with no thought that one's good taste could be different from another's. The most fitting article to use here would therefore be the null article.
Using the flavour sense of taste (definition 1 above, no pun intended) would change the meaning completely. Someone having a good taste refers to them savouring the flavour. So if "she has a really good taste in books", the natural meaning would skew to saying that books record her savouring something.
The humour example follows a similar rationalisation. If you take out the article, the natural reading skews to saying that he has detected strong evidence of humour, analogous to "he has firm sight of land". In this case, though, the "a sense of humour" phrasing is so idiomatic that it overpowers the detect evidence reading.