Great question — answering as the UphillTech team:
It is likely that the textures in the closets is the result of overspray from the texture sprayed on the outer walls and then painted. Often the builder will not bother to fully texture in closets and areas that will not be seen like the walls in open rooms. Recreating the texture will take some experimentation. Some methods that you can try are: Sponging. Dip a sponge in drywall compound and dab it onto the wall. ( Mentioned by brhans in a comment). You may need to try multiple size and types of sponge. Most often used is a sea sponge, but others could be used too. Rag dabbing. Same method of dipping using a rag rolled or wadded up dipped in compound and dabbed to the wall. Brush dabbing....
Full answer by RMDman: https://diy.stackexchange.com/a/331011 (CC BY-SA 4.0).
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