Why are you wasting your time watching TV when you should be wasting your time online?
One botanical definition of an herb refers to "plants without persistent, woody stems. This botanical use of the word ‘herb’ is used to distinguish between trees and shrubs." The banana plant does not have a wooden trunk, even though it can reach a height of 40 feet. The base of the leaves overlap and intertwine in such a way as to form a strong column that supports the plant and eventually its fruit. At the end of each season, the plant dies all the way back to the roots and must begin again almost from scratch.
Alan Davidson, a food know-it-all (in the good sense) and author of The Oxford Companion to Food, says the word herb at its broadest can mean, "a plant whose green parts (usually leaves, sometimes stalks) are used to flavor food." The banana leaf is often used in cooking, most often as a wrap in which foods are cooked, but also to impart flavor. In that context also, the banana (plant) can be considered an herb.
The banana itself is the fruit of the herb - or, in the words of Harold McGee in On Food and Cooking, "a seedless berry of a tree-sized herb related to the grasses... [and then he shows off with some impressive Latin name for the grasses in question]." Already in this question, you've got us talking about herbs, shrubs, trees, leaves, fruits, berries, and grasses - clearly you have to be a botanist to do this job!
And for the record, while a tomato is a fruit botanically, legally in this country, it is a vegetable. Late in the 19th century the United States Supreme Court weighed in on the subject in a case on import tariffs, and decided to cast botany to the wind and classify fruits and vegetables based on how they are generally served and eaten. So the tomato (the defendant in the case), as well as green beans, cucumbers, eggplants, corn kernels, etc. are legally vegetables. (They are not, however, considered to be herbs, even by TV chefs. Bam!)