Do hotels on Phuket allow left storage for luggage? Is it common or anyone have hotel names that do it for sure? Do they charge a fee? I'm starting to move my things there for living there in the future and researched the self storage businesses but feel their too expensive. Any help will be appreciated. Thank you.

rent a thai style apartment 2500thb a month and use as storage?

Nearly all hotels will allow a guest to leave his luggage there for a return a few days/week later or so. But moving here and using a hotel for storage?

Better rent something as suggested above.

rent a thai style apartment 2500thb a month and use as storage?

Which would not be very safe, I think.

No, it's not common. No Hotel or anywhere else wants to be responsible for your possessions. Secured storage isn't cheap here for good reason. Why don't you find where you want to live then bring your things and store them at your house or apartment? But of course they could get stolen there too...