My brother went to Castle Trio. (no correction needed) Don't you mean 'Trio Castle'? We don't say Castle [Name], at least for castles in the UK, with very few exceptions. Let's see: Windsor Castle (the Queen's residence), Balmoral Castle (again, but when she's in Scotland), Warwick Castle, Berry Pomeroy Castle (my local), Caerphilly Castle, Leeds Castle. Ah, but then not far from me there is Castle Drogo.
What about here:
My brother went to the castle of Trio.
Does castle need to be capitalized in second sentence? No, not unless the 'Castle of Trio' is its title, but as I say above most castles are in the form '[Name] Castle' at least in the UK so I'd stick to 'My brother went to Trio Castle', although if Trio was a person and the castle named after him, then one might expect to see 'the Castle of Trio' (but I don't think there are many such castles). Contrast common English usage with French where they write 'Chateau de' eg 'Chateau de Chenonceau'