What type of cable is needed from the fuse box to the stove connection?

For the short distance, that should not be a problem.

However, if you or you guys have so little knowledge that you talk about toggle switches instead of B-16A, etc. circuit breakers and also cannot specify the power consumption of the oven, you definitely should not tinker with the electrical work yourselves!

If you can reduce the maximum power of the hob to a value appropriate for the wiring, you can continue to use the wiring. Do not rely on usage predictions.

The cooktop has a total connection value of 7.8 kW (power of all 4 cooking zones combined). The oven description states: Fuse 13 A. Attached is a photo of the fuse box.

You can't see anything in the photo (or at least I can't). 13A is absolutely no problem with 1.5mm² and the cable length (provided it is properly connected and the cable is not damaged).