A stranger told me he would give me a weekly allowance if I gave him my bank name, account number and routing number. Is that a scam?
Yes, this is a scam. Not because of the information he's asking for, but because of the fact that it's a complete stranger offering to give you a "weekly allowance" for... what? Nothing in return? Nobody just goes around giving money to strangers for no reason. Even official charities have a specific mandate to follow regarding where and how they spend the money they collect.
Sending you pics proves nothing, there are innumerable sources of pics online he could have pulled those from. (You might want to try a reverse image search on one of them just to see what comes up, but please don't take that as any sort of cue to continue interacting with this person.)
Whatever his motives are, you don't want to be on the receiving end. Don't waste your time asking him questions or trying to "figure it out" - just stop responding, block the number / email address, and move on. Rest assured you dodged a bullet, even if you don't know the caliber.