Well, they are kind of both.
A banana "tree" is a herbaceous plant (as opposed to a "tree" is its not "woody"), but the bananas on the tree themselves meet the definition of a fruit - they are what is growing on the plant and normally contain the seeds required for new life (although, in reality, commercially grown banana's seeds are largely useless these days).
Thankfully the last 23 years working as an agronomist in Tully NQ(Australia’s largest banana producing region)has helped me get a handle on what you’re saying,and as you correctly point out modern banana seeds are sterile and that is why TECHNICALLY they are agronomicly identified as Herbs,but thanks for the refresher lol
If it grows on a tree, it's a fruit. A fruit is something that grows from a plant in order to be consumed, ie, strawberries (grow from patches), pumpkin (grow from patches), oranges, apples, peaches (all coming from trees), tomatoes, cucumber are ALL fruits.
Veggies grow as the are and do not evolve from a plant and are eaten from what they grow out of, eg, potatoes, carrots and lettuce are all veggies.