Sometimes. It depends on the wound, so it isn’t a general yes.
During a wet-to-dry dressing change, the cleaning step is: rinse your wound with water, gently pat it dry with a clean towel, and do not rub it dry. In some cases, you can even rinse the wound while showering.
In some cases. Which cases depends on the wound, so it’s a question for whoever is managing it, not something to guess.
After surgery, it works the same way. Your surgeon will let you know when it is OK to shower or bathe after surgery, and usually it is fine to shower after 24 hours. If your surgeon gives the OK, gently rinse the wound with water as you bathe, and do not rub or scrub the wound.
The firm rule is about soaking, not rinsing. Showers are better than baths because the wound does not soak in the water. Soaking the wound could cause it to reopen or become infected. A shower runs over a wound; a bath sits on it.
What to do afterward is the part most often skipped. Gently pat dry the area around the wound with a clean towel, let the wound air dry, and apply a new dressing. Do not use lotions, powders, cosmetics, or any other skin care products on the wound.
If you have been given different instructions for your own wound, including a waterproof dressing to leave in place, or a plastic bag to cover it with, those came from someone who could see it, and they come first.