If the tiles are porcelain, they can not be porous, just rough. Sounds like the type of tiles that should not be used in a bath/shower, bad selection. The tiles are probably glazed, but it's a matte glaze. Glazed tiles do not have to be shiny.
The tile looked nice and smooth in store but not so much in our house.
How is that possible? Maybe you got a different tile?
Tiles look dirty all the time and so does grout.
Sounds like a maintenance issue to me. Do you not wipe the walls and floor after use? Ever?
Tile so rough that it shreds mop head apart.
A mop? You don't use mops to clean bathroom floors. You gotta get down.:laughing:
Two ways to proceed without ripping all the tiles and starting from scratch.
Use a tile & grout cleaner. Then seal with a good penetrating sealer which will help keep moisture from absorbing into the grout and fill the microscopic pores of the tiles whether glazed or not. Or....
Clean as above, then apply a topical finish to give the tiles a gloss. They're available in varying gloss levels. But, this method means you'll have to keep applying the finish on a regular schedule to maintain the "look". I do not recommend this method, but you can get a gloss.