OED says:
herb: 1. A plant of which the stem does not become woody and persistent (as in a shrub or a tree), but remains more or less soft and succulent, and dies down to the ground (or entirely) after flowering. 2. Applied to plants of which the leaves, or stem and leaves, are used for food or medicine, or in some way for their scent or flavour.
The banana seems to be like 1, but not so much like 2.
It says this of fruit: The edible product of a plant or tree, consisting of the seed and its envelope, esp. the latter when it is of a juicy pulpy nature, as in the apple, orange, plum, etc.
That sounds much more like a banana which it calls a fruit. Because the leaves are not used as an aromatic in cooking, it does not seem to be much of an herb. So, lacking further evidence to the contrary, I would have to disagree with banana.com and say that it is a fruit.