I'm presuming here that your service is three-phase "wye" with 230V between each phase and neutral. First, let me say on behalf of all Americans how jealous we are.
I can't see the Ikea link, but I'm guessing the cooktop is listed (certified) to be connected to two separate phases like this. It's common for EU homes to be served by two phases; getting all three is kinda nice.
I don't know German Code but from a North American perspective... Why change the breaker? You need 16A circuit protection on each leg, and you have it. Connect the stove to two legs, and the oven to the third. Done.
Now you have what is called, in North American parlance, a "multi-wire branch circuit" or MWBC.
Yes, all sides of the circuit will trip together. That's mandatory in the USA for a good reason: to protect repair personnel. The breaker functions as a shut-off switch, and the rule is the shut-off switch must shut off all legs at once. That way a repairman isn't caught unaware that part of the MWBC is still energized.
The 13A requirement for the oven is a peculiar number. 13A is a commonly used fuse or circuit breaker value in the UK. I don't know if the oven will be alright with a 16A circuit breaker, I'd check with the manufacturer. I would imagine so, since they engineered it to sell to all of Europe.