Well, I sure as hell didn't understand the new meaning behind the Belmonts hunting down evil/Dracula in this game. Unless he was referring to Gabe's quest to save Marie, then that's pretty stupid.
Makes no sense in the game as it is; could make a lot of sense in the perspective of Belmonts chasing Gabriel, tho.
In short, LoS extabilishes a "blood" tie between Dracula and the Belmonts. How does this pan out, we don't know. 3 theories:
1) Gabe has a son between being resurrected as an angelic being and becoming a vampire. This would explain the "power" of the bloodline (as they would descent from someone infused with holy power) and also why they are tasked to chase Dracula specifically (clear up the family name/destined/cursed to do it)
2) someone unrelated to Gabriel takes up the Belmont name to fight Dracula. He may do so because he's an orphan and does know the legend of Dracula's origins and sees it as fine irony ("Good, now this is personal, and I'm gonna pick up your name so that when I meet you we can have a laugh") or because this guy goes through something that bonds him and his bloodline to Dracula ("so you have forged the whip with meteoric iron and your blood, and your family will be able to hold its power. How should I call you?" "Mmmh.... Belmont. Call me Belmont. The sucker will understand."). And so on.
3) there IS a Belmont family in the world. It doesn't have to necessarily be a family of monster hunters. Actually, it shouldn't. Butch Belmont and his wife Cybill are a couple of dairy farmers leading a peaceful life in the country. One day Dracula's goons storm the farmstead, killing them both, kidnapping their teenage daughter to be Drac's next wife and leaving 8 years old Christopher for dead.
But he's not.
So he grows up and becomes a monster hunter for obvious reasons, and there you have it, the Belmont family. I wouldn't see it as "redicolous coincidence". If played well, it could be ironic destiny. It could play well into LoS' "christian" themes. Gabe would certainly see it as God's own toying with him.