Yes you can wash clothes in the shower.
Many people washed clothes for years before washing machines were invented.
If your shower has a bathtub you can fill the bathtub up with some water and then add your laundry detergent.
Then place the clothes in the bathtub and use your hand or something else to stir the clothes in the water to agitate them.
Then pick the clothes up and use your hands to twist the clothes to wring them out and then use the shower to run water over the clothes to rinse the laundry detergent out of the clothes.
If your shower has no bathtub then you can use a bucket to dip the clothes in and agitate them around and then use the shower water to rinse the laundry detergent out.
I used my bathtub in an apartment to wash clothes before when I had no washing machine and didn't want to deal with the laundromat.