They're really just a protocol. You can implement them in various ways. They will always be some sort of delimited continuations but a "function call" or continuation passing style or anything of the sort does not have to be involved at all.

For example, let us say I don't allow "multi-shot" continuations like in your library, and I'm implementing Algebraic Effects in my own interpreted language.

One way I can implement effects and handlers is to have a handlers get registered in a stack, then when an effect is triggered, save the IP and current stackframe, search for the right handler and jump to it. "resume" then just resets the stackframe, pushes a value into the stack, and sets the saved IP.

(Only saw your edit after posting, sorry, but yes)