'Is always biting', suggesting a (possibly annoying) habit, is what we would almost certainly expect here. However, 'always bites' is not a mistake.
The use of always, normally associated by virtue of its meaning with a simple tense seems at first sight illogical. However, as the use of the Progressive form with a short action stresses the repetition of that action. The combination of the progressive form and always tells us that this is a an situation that actualises repeatedly, but because the duration of the whole series of biting is limited, it is not presented as a permanent state of affairs
This combination is associated by some writers with some idea of the speaker's emotional attitude, but this will be made explicit (as it is in the test example) not just by the words, but by the whole context of situation and the speaker's tone. It is not true to suggest that it always expresses the speaker's irritation; with change of tone of voice and facial expression, the person saying such words could express irritation, resignation, amusement or a number of other feelings. Here, as is almost always the case in English, it is context and other factors that express feelings, not simply the words. The combination can just as easily be used to express pleasure, as in:
He's always buying me flowers.
