(Building on a ton of comments)
Your cooktop needs two hots + neutral + ground.
Your supply cable has two hots + ground. That is a problem.
The cooktop instructions allow for two hots + neutral, using the neutral for neutral + ground together. This is supposed to be only for old installations (i.e., grandfathered), which is (a) not recommended for safety reasons and (b) likely not allowed in your case anyway. Not an option.
There is, often, an allowed fix (again, supposed for old installations only, but it is safe). That is to retrofit a ground from elsewhere. But that only works if you have two hots + neutral and are missing ground. But you have two hots + ground, and that means the third wire is not insulated (and if it were insulated, it would be green, which is a ground-only color). Not an option.
Unlike ground, you can't pull a neutral from another circuit. That would run the risk of overloading that neutral - and neutrals are not protected by breakers like hots are. Not an option.
I only see 2 options:
Switch for a 240 V only cooktop.
Run a new cable that includes two hots, neutral and ground. Maybe you can run a cable the ugly way - up to the ceiling (i.e., through cabinets) and along the ceiling around to wherever the breakers are? Or use the old cable (depending on how it was installed) to pull a new cable in to replace it.