As an addition to Greendrake's answer:
A lot will depend on the exact nature of the agreements, both the original lease agreement, and any agreement around the repair of the washing machine.
As written in Greendrake's answer, there is no groundreason to believe the tenant can claim ownership of the washing machine. The machine was replaced under a guarantee that was extended to the owner of the original machine, so the replacment is theirs, too.
However, the tenant may be able to claim a (partial) refund of the money they paid for the repairs:
- They could claim that they paid the money believing that the owner would refund them. If they could show communication implying this (such as the owner writing "Go ahead, use the guarantee, that's what it is for"), that would help.
- They could claim that they acted in the best interest of the owner, believing they would have paid if they could (because $100 for a machine is a great deal). In law this is referred to as negotiorum gestio - essentially the tenant is sponaneously acting in the best interest of the owner, since the owner is not available. Again, the tenant would need to show that it was in the owner's interest to pay.
If the case should go to arbitration or court, the tenant could make the above arguments, and the court would need to weigh them. The court might then decide that the owner needs to reimburse the deductible. The reimbursement might only be partial, because even if the payment was in the owner's interest, it was also in the tenant's interest. That would depend on how the court weighs the competing interests in play.