water filters also have the incompatible proprietary filters problem. I wanted to try filtering some water so I could make coffee that wasn't bitter but every water filter seemed to want me to buy a bunch of plastic to go around a replaceable filter. In the end I settled on pyrolysized cherry wood branches, (activated carbon, or biochar). It seems them carbonated bamboo is also used in tea making. It worked but it certainly took a long time leaving the sticks in the water.
In the spirit of DIY/OSS here, I think ideally you’d want to emulate something like a Berkey water filter, these are basically just steel tanks with large threaded cylindrical charcoal filters (2” diameter, ~10” long) that use gravity to pull the water from the top to the bottom.
A simple 3D printed housing filled with some charcoal/carbon media to trickle filter like a Brita would be interesting to see its effectiveness.
At my parents house, we set up a whole-house water filtration system by repurposing water softeners. empty the guts of one, refill with aquarium supply activated carbon, then adjust the electronics to your liking and you're good to go. Those systems are designed to run 24/7 and soften/filter all the water going into your house.
There are standard sizes and housings for water filters. No fridge uses anything standard, but you can find them. you need some sort of plastic (or at least something food safe) around the filter though.