As far as I know, there is no 'look-alike' or 'feel-alike' when it comes to asbestos shingles. I'm going to go out on a limb and see if I know where you're coming from.
These shingles are classified as Non-Friable. In other words, the hazard becomes readily apparent if you inhale 'dust' from the breaking of the shingle. As long as you handle them with gloves, you're fairly safe to do so. Just don't go smashing them up and such. And if you do, be sure you're UPWIND and no one else is 'around' you to breathe in the dust. In very, very small quantities, many municipalities allow disposal in with your Household Waste (or in the trash.)
This particular siding material was marketed from a fire-protection angle. Plus, it was very inexpensive to make and install. Unless you want to re-side with a completely different material, you're probably better off just spending the $$$ and buying a bundle at the Home Center. If you're lucky, you may be able to get them pre-painted in Tan or Blue (as well as White.) Home Depot has them in my area. Try the other Home Centers. If you're still having trouble, check with Siding Companies in your area.