Yes, but...
Yes, this does mean that it is possible to gain feats without qualifying for them with levels in a class. But this is not intended to be a loophole to get infinite feats. On the contrary, but it should be a rare occurrence, and a reward for pleasing someone powerful by doing something remarkable.
As the first part of the Training section you quote says:
This kind of training isn't widely available and thus is highly desirable.
And as the introduction to Other Rewards on DMG 227 says:
This section presents a variety of ways that gods, monarchs, and other beings of power might recognize the characters' accomplishments, including supernatural gifts that give characters new capabilities; titles, lands, and other marks of prestige...