I know this is an old thread, but I figured I should mention that I used to use running shoes 20 years ago to play paddleball (same shoe requirements as tennis) and had no problems, BUT it's true that running shoes are made differently in every way and the rubber is less dense.
As for whether tennis shoes are better for your ankle, that's the common belief and I'll go along with that, but I have to admit that I'm not 100% convinced. Tennis shoes provide more ankle support, meaning that they make it harder to rock your ankle sideways, but they also provide less padding than running shoes which puts more pressure on your joints, tendons, etc. with every step. And when a tennis shoe tilts to a certain point, you're teetering on the rounded edge of the shoe, but with a running shoe the sole may still be flat on the ground.
And remember, there are cross trainers. If you're bold enough to choose a shoe that's not made for your sport, then go with cross trainers. They're less stiff and more padded than tennis shoes, but they're more durable, supportive, and have more traction than running shoes. That's what I'm considering buying. I'm even looking for a more flexible than average cross trainer, but I won't take a chance on running shoes any more.
I think the cross trainer that's most like a running shoe is the Nike Air Edge Essential. Unfortunately, the one that's part red didn't look good in person and I can't find the other colors in stores. I may mail order it.