Yes you will have fishy chicken. In steam ovens the moisture will carry flavors in most cases. Not quite like a standard convection oven.

are you speaking from real life experience? according to a lot of marketing materials, one of the advantages of steam oven is that there is no cross-over of taste or smell if you cook 2 different dishes together. so who is right?

Supposedly the flavors don't transfer, that's what the manufacturers say. I have reheated several things at once, like say pizza and eggs (don't judge), without any noticeable issues. I have yet to cook raw fish and chicken at the same time though.

I have a Wolf CSO and I have cooked fish and poultry in separate trays at the same time and there is no transfer of flavors.