If showers are not a part of your routine, then, when showering, your mind will be preoccupied with things like "how to shower," "how not to slip," "how not to get burned with hot water," "how to control water temperature," etc., and will not be able to achieve the state of relaxation necessary for generation of new ideas.

The key is that the activity has to be both relaxing and routine (double "r"...). For you, it's bathing, for others it's showering, yet for others it's walking in the woods. If the activity is not relaxing, you will be preoccupied with how to control it and with resulting stress. If the activity is not routine, you will be focused on its novelty rather than letting your mind wander on its own.

Maybe subconsciously it's more stressful for me to have a shower, but I have far more showers than baths, and certainly I don't need to really think about any aspects of it, consciously.

It's just that in a shower I never decide to relax and enjoy it, it's always a method of cleaning, or of waking up, or of heating up, not an activity for the sake of enjoyment.

The shower is utterly routine for me, but I also never get ideas there. It's sad - I feel like I'm missing out on something.

My best idea generator is the walk away from my desk & to my car at the end of the day. I have solved so many bugs immediately after exiting the office building...

My theory is that it's the white noise sound from the water that helps concentration.