Is the elder brother entitled to own part of the taxi that was registered in his name, even though the younger brother is the one who bought it with his own money, and the elder brother is seeking to sell it?

Ownership is legally established for its proprietor, and it is not permissible to dispute it. The essence of contracts lies in their meanings, not their mere wording. Thus, official documents do not alter the ruling concerning the true owner if they contradict reality. The shop, car, and apartment mentioned in the question are the properties of the younger brother. His ownership is not relinquished merely by writing a part of them in the name of his elder brother for the sake of reverence, not for transfer of ownership. Therefore, it is not permissible for the elder brother to dispose of them or to dispute his brother. If the elder brother's assistance in the shop was voluntary, he is not entitled to wages. If it was a lease and the wage was not specified, he is entitled to the wage of a similar service (ujrat al-mithl), from which whatever he took from the shop should be deducted. It is recommended to resolve the dispute between them to preserve familial ties, and the younger brother is advised to be gentle in reclaiming his right and to seek assistance from Allah, the Almighty.