Is it permissible for a garage owner to benefit from damaged car parts left by customers, by selling or exchanging them for his personal gain, given that the customer does not benefit from them at all?

There is no harm in taking damaged parts that their owners have left behind, because they have relinquished their right of ownership over them. Thus, they become permissible for whoever takes possession of them first, and it is permissible to benefit from them through sale or otherwise, as stated in Al-Insaf: "Whoever is first to a permissible thing, such as game, ambergris, musk... and what people discard due to their disinterest in it, then he has more right to it." However, you have no claim against the manufacturing company.