Mate, I’ll be honest with you here - I grow bananas on my property and I can definitely tell you the last term I’d use to describe their trunks is ‘woody’. They’re so moisture-laden and ultra porous that anyone who’s ever had to cut or cull banana will know for sure that they’re not made of wood. You can easily carve through a 15-20cm trunk of a banana plant with a sharp machete and one strong swing - try that on anything generally considered to be a tree and you’ll be lucky to get a fair chip out of the trunk.
I’ve got no skin in the game as to whether or not they’re trees, or what the fuck a ‘tree’ even is, but anyone who’s dealt with growing bananas is pretty unlikely to consider them in the same group as woody trees. Damn things grow like weeds anyway!