Well, no. There is plenty of room to debate definitional questions around open source. For example, is open source even the right goal, or should we be aiming for something different, like free software? Is the GPLv3 a step forward from GPLv2, or not a great idea? Wherever you come down on those questions, I think virtually everyone involved with open source or free software would agree that part of the appeal of such software is that you don't have to pay license fees to use it.
Relevantly, it is also not clear whether or not MongoDB's license will meet with the approval of the Open Source Initiative (see relevant email threads about SSPL v1 and SSPL v2). MongoDB is trying to claim that their new license is open source, but if OSI rejects their new license text as incompatible with the Open Source Definition, a lot of people are going to feel that it isn't open source regardless of what MongoDB cares to claim. It's particularly hard to view Amazon's move as an attack on open source when it is no longer even clear that MongoDB is open source. If this move is an attack on anyone, it's an attack on MongoDB, not on open source.
To be clear, I'm not saying here that Amazon isn't attacking open source in general, or that they are, just that this particular claim - which I'm inclined to attribute as much to MongoDB as to TechCrunch - is misleading. If you're looking for a database which is unambiguously open source and, for that matter, not controlled by any single company large or small, you might want to ignore both DocumentDB and MongoDB and check out PostgreSQL, whose license is indeed OSI-approved (more info), and which has just won the DB-Engines Database of the Year award for the second year in a row.