I

s a banana a fruit or a herb?

Both. A banana (the yellow thing you peel and eat) is undoubtedly a fruit

 (containing the seeds of the plant: see 'Is a tomato a fruit or a vegetable?),

 though since commercially grown banana plants are sterile, 

the seeds are reduced to little specks. The banana plant is called a 'banana tree' in popular use, but it's technically regarded as a herbaceous plant (or 'herb'),

 not a tree, because the stem does not contain true woody tissue.