Is it a condition for the validity of a transaction to own the object of the transaction and to specify it? And is this transaction permissible (halal) or forbidden (haram), considering that a share in a shop was bought for an indefinite period until the financial hardship is removed, on the condition that it be bought back at the price determined by the buyer?

This sale is invalid due to ignorance of the sold item (al-majhūl). One of the conditions for a valid sale is that the sold item must be known in quantity, type, and kind at the time of the contract. This is because such ignorance leads to gharar (excessive uncertainty or risk), which is forbidden in Islamic law. This is a great principle among the principles of sales that renders the sale invalid. Therefore, the questioner is entitled to receive back from the other party what he paid him. If they wish to conclude a valid sale contract, he should specify the share he will give him in exchange for his money. As for his promise to buy it from him in the future, there is no harm in it, as long as it is not made as a condition or obligation.