I live in a GCC country which is more conservative than the UAE and I would definitely say that breastfeeding in public is more acceptable here than it is even in England. You won't get hassled for doing it, at least not by the locals. Also, there are more places for women to breastfeed (every mall etc has a women's prayer room where even non-Muslims can go and do a feed), there are nursing rooms in all the baby shops, and there isn't a problem with feeding in cafes etc as long as you are reasonably discreet.
This all boils down to the fact that the Qu'uran states that it is the RIGHT of all infants to be breastfed until they are two, or earlier if they naturally wean before then.
But, one of the big problems health professionals are facing here is that breastfeeding rates are incredibly low, especially among the local population. This has a lot to do with the fact that everyone has a maid (local families nearly always have one maid per child) and the maid does all the feeding, so most kids nowadays here are raised on formula.
There is a lot of chin-scratching going on about how to change this, and I guess this law has come about as a result. It's not the ideal way to do it (understatement!), and it will get largely ignored and won't be enforced I am sure, but that's the nature of life here, they often come up with daft rules to try and tackle social problems unsuccessfully.