You can buy a pod for the sensor that attaches to the laces on any type of shoe. It just works best with shoes designed with Nike + as the sensor sits under the insole out the way.

can only be used with Nike

As far as I'm aware it will only work with some of the Nike running shoes.

You can buy pouches to put the sensor in that attach to your laces. I had mine rigged up to Saucony shoes.

The sensor can be used with any trainer. For example, I have Nike+ trainers, but when I do trail runs I use it in my Salomon Speed Cross 2. I tuck the sensor in the tongue where the lace is stored.

As long as the sensor is moving it will work, Don't know much about Asics, but you could probably tuck it in your trainer as long as it was comfortable to run with.

It can work with any runners, but you'll need the shoelace pouch.

The sensor would normally fit inside the sole of the shoe.

It works with any shoes from other brands and even with any shoes from Nike which is not 'Nike plus ready', using a small case to put the sensor on your shoe strap, like Marware Sportsuit.
'Nike plus ready' shoes have a pit to install the sensor under left inner sole, and this is only the difference.

I have iPhone 3G and work perfect with Nike+ (I had to download from app store)

Yes, with iPod 2G+ and iPhone 3G+. Best choice.