You don't need tennis shoes to play tennis however tennis shoes are more comfortable to wear when playing tennis.
Professional tennis players play tennis while wearing tennis shoes because it really helps their tennis game but for the average player that just wants to play tennis you can play tennis in your regular running shoes or whatever shoes you want to wear.
I've worn tennis shoes as a teen and kid and never played any tennis before and I thought as a kid they were just regular shoes with that name.
However later on as an adult I found out that tennis shoes were made different for people playing tennis.
Regular running shoes will work just fine when playing tennis but if you're playing tennis professionally I recommend wearing tennis shoes.