The shoe industry would like you to believe you need a different shoe for every activity. I was once asked in a Rockport outlet if I was looking for "walking shoes." I asked the salesman if they sold any shoes that would be inappropriate for walking.
I've played tennis at least twice in running shoes because they were all I had with me, and things went fine. I'm not altogether convinced I was in peril playing in them and suspect my ankles do more to keep me from rolling over than any shoe I've ever worn. I've seen people roll their ankles countless times in tennis shoes, not at all clear they afford greater protection, and the cushioning of a running shoe perhaps would afford more protection against developing something like plantar fascitis. Johnny Mac, some will recall, played in cross trainers, sort of a hybrid running and general purpose shoe, for a good portion of his career.
"Where's the REST of me?"
Ronald Reagan, in "Kings Row"