Is 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 answer regarding tomatoes), though since commercially grown banana plants are sterile, the seeds are reduced to little specks. However, the banana plant, though it is called a 'banana-tree' in popular usage, is technically regarded as a herbaceous plant (or 'herb'), not a tree, because the stem does not contain true woody tissue.