I rather doubt there is anything wrong with the car; the tires are getting on in age in that they're probably four or more years old, and the rubber, already relatively hard, is getting harder. You could certainly upgrade your tires to sticky summer tires, but you'd have to live with the trade-off of shorter tire life (like less than 20,000 miles).

The thing is, your Mazda5 probably weighs roughly 3,500 pounds, and that's a bit on the heavy side for a car shod with only 205 section width tires; I suppose you could go to a 225/45 R17 sized tire, however, you may have a tire rubbing issue when the steering wheel is turned to full lock.

Tire spin
 

I have 2013 Mazda 5 and the front tires spin. I am pretty sure it is a defect and that's why the model was discontinued. I shift the transmission to manual model and start at second gear. It helps a bit, but yes the hills are tough. I am thinking of buying two winter tires for the front, lower the psi, so it is extra sticky, and see if that works. I don't mind it too much but, yes, my wife can't handle that.

Tire spin
 

And, yes, many Mazda 5 owners have complained about this.