Here is the replacement cartridge. Definitely a screw holding the handle.
Repair your leaky faucet with the DANCO DL-10 Cartridge for Delta Scald-Guard Tub/Shower Single-Lever Faucets. Replacing a faucet cartridge is an economical alternative to replacing the entire faucet fixture and a dripping faucet wastes both water and energy. The installation of a cartridge will...
www.homedepot.com
Here is a handle. If the handle rotates and the handle travels vertically to open the faucet, it might confim my thoughts that that is not the original handle.
www.outdoorsupplyhardware.com