I actually think brick and mortar stores
are competing with Amazon - and there is also an emerging hybrid approach that threatens Amazon as well.
In many places, suburbs and cities alike, people don't want stuff delivered to their doorstep. The existence of Amazon Lockers is proof enough of this - sotres now offer online shopping with local pickup (Best Buy, Staples, and IIRC Wal-Mart). This competes with Amazon.
Wal-Mart is also attempting to reposition itself as an Amazon alternative (i.e., pure online shopping) and has the capital and existing supply chain infrastructure to do so (they are one of very few).
On top of that Amazon continues to struggle with specialty goods. Amazon is very, very effective at selling mass-market products, but their attempts to move upmarket have been extremely mixed. Lower-volume, premium goods are not their forte, and brick and mortar stores (as well as niche online shops) still have a firm grip on that.
Not to mention we know that the public is extremely price-sensitive. Amazon continues to price-compete with Wal-Mart, despite its substantially lower volume, leading to even thinner margins. They do this out of necessity, because at this point for many people the conveniences of online do not trump a quick trip to Wal-Mart if the product is cheaper.
I don't think it's at all prudent to discount pure brick and mortar businesses as competitors to Amazon. Amazon very much exists in the same sphere as all retailers, online and off.