Whenever you take out a loan for a car/bike, then if the person (brother) does not have good credit or not enough credit, then the lender will require a co-signor on the loan (dad). Now, (if) the payments are not made by brother, then your dad would be required to pay it. Now, once the bike is paid off, then the lender would send a release of the bike being paid off, then the owner (brother) and (maybe) dad would then put the bike into their names as the owner of the bike. Many times co-signors are not listed on the title, but in some cases they are.
Now the lender does not care (who) makes the payments, since all they care about is getting their money till paid off. Now if boyfriend stopped making payments, then the lender (can't) go after boyfriend, since boyfriend is NOT on the loan/contract, but would go after brother or your dad to make the payments.
Your boyfriend has NO legal right to this bike. Whoever is listed on the title is the owner.
Boyfriend (gave) the bike to the brother as a gift. Boyfriend paid for the bike and it appears that he never asked for his money to be repaid back.
When does a gift become a loan or an owner? Even if boyfriend went to court and proved he made all the payments, then the judge would still say it is a gift, since (unless) there is some written agreement that brother agreed to pay for the bike, like boyfriend saying, where is the money you owe me, or if brother says (I will agree to pay you back), then if nothing that brother agreed to pay anything, then once again it is a gift.
A gift is gift and nowhere in your question, does it "imply" otherwise.