I'm trying to rename my business and I thought of adding studio to the name instead of productions. I have seen some companies add studio when they do not have a "studio" What do you guys think? Do you need a studio to have studio in your name?

If you call yourself a studio, you should be prepared to say YES when someone asks...

"Can we meet at your studio?"
"Can we shoot the interview at the studio?"
"Can I pick up the footage at the studio?"
"Can I get a tour of the studio?"
"Can we webcast from the studio?"
etc.

And if you call yourself "Studios", you should be prepared to show them the other studio too... my 2 cents.

Christian said it perfectly.
I'd add that if you have a single location with multiple "shooting rooms", you could use the plural.