After the ending, the Hero (and his bloodline) basically ends up ruling over half of the world anyway.
In the original release of Dragon Quest, the message on Erdrick's tombstone was in the same kind of Ye Olde Butcherede Englishe as the rest of Tantegel, but in subsequent versions, it notably has much more contemporary speech in stark contrast to everyone else. There's a reason for this: the hero Erdrick didn't come from this underdeveloped world.
In-game dialogue from various NPCs suggest a justification of the With This Herring trope present: you're not the first random adventurer to claim lineage from the Legendary Hero—in fact, you're hardly the most impressive-looking claimant of the lot. Some of the more skeptical don't even believe you until you're slain the dragonlord.
Somewhat lessened in remakes, where the princess's cell is rather nicely furnished. With that being said, chances are she's being kept fed by her captor(s).