If a deaf person had tinnitus, they'd probably never recognize it

Nope

https://www.ndcs.org.uk/information-and-support/childhood-deafness/additional-needs/deaf-children-and-tinnitus/

Congenital? No. Acquired? Yes. The area of the brain that processes and interprets sound has to develop. Without sound input as a child, that won't happen.

Current leading theory of tinnitus is called the 'central gain' theory. This is where the brain becomes accustomed to seeing signals from the ear at a certain level, and when that neural level is no longer at that level it will add in its own noise to make up the difference. This noise is then perceived as a tone or sometimes a broadband sound, commonly described as either a ringing or a whooshing sound. Sometimes it can also be described as crickets. Depends on the person and cause. Not all hearing loss comes with tinnitus, but most tinnitus comes with hearing loss. In audiology school we had a whole class on tinnitus and covered many interesting aspects exactly like your shower thought here and went over papers on every angle you could think of. It was fun. But in the end, the brain has to at a minimum know what sound is to even perceive sound.

Many years ago I did a charity event that involved a lot of camping. If you went solo (as I did) you had to share a tent with another solo someone. The guy I tented with was deaf. At night the campground was super noisy and it was hard for me to get any sleep. He said, "This is my super power....I just take out my hearing aids and noise goes away. Really useful if your partner snores."

Unfortunately tinnitus often accompanies sudden deafness.

ethical edit: For a toss-off gag that even I thought was a bit sketch, I'm learning a lot about this situation and I appreciate it