Is it permissible for one who loaned money to his company to buy shares in it equal to the value of his loan, based on its value a year ago, as compensation for the delayed repayment of the loan, relying on the story of the Companions of the Cave?

It is permissible to convert the loan into company shares at the price of the day of conversion. It is not permissible to do so for less than that, because the difference is considered usury (riba) and a profit from something that was not under your guarantee. The Prophet, peace and blessings be upon him, forbade this when he said to Ibn Umar: "There is no harm in you taking them at their price on that day, as long as you do not separate while there is still something between you." As for using the hadith of the three men in the cave as evidence, it is out of place, because what the man did was a kindness and a voluntary act from the debtor, not a stipulated exchange from the lender. In conclusion: You are not entitled to demand any increase in return for the delay of the loan, whether in cash or shares at a lower price. The reward of a loan is better than any worldly profit.