Likely not a popular opinion, but if running shoes work for you, keep using them. I have many friends that find better cushion and comfort playing in running shoes than tennis shoes.
You might need to go to tennis shoes depending on your specific case, for instance:
1. do you burn through your outsoles very quick? I do, and this is the reason I moved to tennis shoes.
2. do you break your uppers because you drag your feet when you play tennis? Then you might need tennis shoes with upper protection.
3. do you like to slide more or less than with your current shoes? Different brands of tennis shoes give you more options for sliding.
4. do you roll your ankles? Then you need shoes with more lateral stability. Tennis or basketball.
5. do you get pain in your knees? Then you need shoes and/or insoles with better cushioning and shock absorption.
etc
Some people choose their shoes just because they like the color - just saying