This article says Amazon accounts for $1B of UPS 60B revenue -- only 1.67%[0]. And Amazon cut out almost all of the profits from those deliveries, with discounts over 70%... so Amazon contributed little to their income.
Wasn't able to find numbers for Fedex.. but they say no single customer accounts for more than 3 percent of revenue[1]. And it would be safe to assume that they negotiated just as hard with Fedex as UPS.
USPS might actually be subsidizing Amazon's packages because of how their costs are calculated[2].
So not so bad for the other carriers.. and unless Amazon opens it up to other companies to use, they aren't going to rival UPS or even FedEx in revenue.
0. https://www.thestreet.com/story/13433347/2/59-billion-reason...
1. https://www.cnbc.com/2017/10/05/amazons-new-delivery-program...
2. http://fortune.com/2017/07/16/amazon-postal-service-subsidy/
Edit: since the first link was written... Amazon has doubled their shipping expenses to $16B (of course, some of that cost is setting up their own shipping service): https://www.sec.gov/Archives/edgar/data/1018724/000101872417... If they can get that entirely under their service, that'll be significant.. but for comparison, Fedex has $50B/yr revenue.