Thaumaturgy as we know it is an Ul'dahn philosophy, so being a native citizen of Ul'dah puts him in prime position to become one. Thaumaturges aren't simply mages that throw fireballs. They're priests of Nald'thal (excluding adventurers who have just gone there to learn to throw fireballs). So depending on what you want to do with him, he could have joined the Nald'thal priesthood at a young age. You say he wants power, which is the what the Ul'dahn philosophy is almost entirely based around. If you want more info on the Order of Nald'thal, you can check out Sounsyy's compilation in the lore discussion section.
Edit: Just adding real quick that it doesn't matter much if your character is poor or an orphan. Money is everything in Ul'dah, but your lot in life isn't determined by how much you have at birth. It's a place where anyone can potentially make something of themselves. Two members of the Syndicate are entirely self-made. Lolorito, who began as a simple porter in another company, and Raubahn who's a refugee previously imprisoned on suspicion of being a Garlean spy. If they can make it to the Syndicate, Cain can easily become a thaumaturge in any number of ways.
Edit: Just adding real quick that it doesn't matter much if your character is poor or an orphan. Money is everything in Ul'dah, but your lot in life isn't determined by how much you have at birth. It's a place where anyone can potentially make something of themselves. Two members of the Syndicate are entirely self-made. Lolorito, who began as a simple porter in another company, and Raubahn who's a refugee previously imprisoned on suspicion of being a Garlean spy. If they can make it to the Syndicate, Cain can easily become a thaumaturge in any number of ways.
