Should I be able to sing each note of the melodic interval in order to recognize it?
You don't need to be able to sing the interval in order to recognize it, but it's still a great idea to do so! Your voice is the best device you have available in ear training, since it's the more direct way to get out of the abstract world and experience the interval as a literal experience.
At least for me, the most effective and efficient ear training exercises involve singing in some way or form.
Or... Should I be able to recognize them only by their unique flavour?
You should be able to do both, and more. An interval can be seen and identified from multiple perspectives, and while it's useful to isolate one of those perspectives for training, in the actual practice you are using all those perspectives at the same time.
More concretely, while listening to melodic intervals focus on:
Discern one note from the other. If you already internalized the interval melodically, separating one note from the other makes the recognition somewhat similar to melodic intervals. Harmonic interval exercises are as much about identifying qualities as about leaning to "see" where one note ends and the other begins.
You should also be able to discern the interval by its "flavor" or quality, without necessarily thinking in melodic terms. For example, minor seconds and major sevenths have a very distinctive beating because of the semitone distance. Major seconds and minor sevenths still have a beating, but much reduced in intensity (beatings in general are more noticeable in the lower registers). Perfect intervals (4ths, 5ths, 8ves) are ridiculously stable. Thirds basically sound as their respective triads. And so on.
How to recognize harmonic intervals?
In short: You should be able to separate the notes in your head and identify them in a similar way to melodic interval recognition, but you should also be able to identify them by their quality or "flavor" without thinking in melodic therms. In practice, you are using both perspectives at the same time.
Depending on your background, it's possible that you'll find one perspective much easier to apply that the other. That's perfectly normal and don't worry too much about it. Just try to apply both in your training and practice, and with time both will get in a good form.