All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger.
The owners of mechanic shops and the like of service providers are liable for any damages caused by mistake on their part or their workers' part in the course of their work because the owner of the mechanic shop in this case is “Ajeer Mushtarak,” namely a worker who offers his services to many clients, and he is liable under Islamic law for the damage caused to the property in his possession regardless of whether it is caused by his own actions or his workers' and whether it was done on account of negligence or not. This scholarly view is adopted by the majority of the jurists. Hence, this mechanic is liable for the damage caused to your car by his workers.
Kashshaaf Al-Qinaa’ (a Hanbali book) reads:
“If the owner of a dressmaking shop was given a garment to be sewn and he assigned the task to his worker who then tore or damaged it without any transgression or negligence on his part, the worker is not liable for the damage because he is an Ajeer Khaas (hired-worker contracted by one party to perform a specific task in a pre-defined amount of time), and the business owner (owner of the shop) bears liability for the damage because he is an Ajeer Mushtarak." [End of quote]
Ghamz Al-ʻUyoon wal-Basaa'ir (a Hanafi book) reads, "If a man gave a cloth to the tailor to shorten it for a fixed fee and the tailor assigned the task to one of his workers who tore it, then the tailor is liable for the damage and not the worker ... The reason why the tailor is the one liable for the damage is that the worker works for him with his permission. The same ruling was asserted in Al-Walwaalijiyyah (Fataawa of ʻAbd Ar-Rasheed Al-Walwaaliji).” [End of quote]
Hence, the owner of the mechanic shop is liable for the damage to your car because of the mistake on his part or on part of his workers.
However, this does not waive his right to the agreed-upon wages if he did do the service that you requested from him. Therefore, you are obliged to pay him his due wages. You are also entitled to demand compensation for the damage caused to your car tire.
If the two of you reach a mutual understanding and agree that each of you would give up his right in favor of the other; i.e. the mechanic gives up his wages in return for you giving up the compensation for the damage caused to your car, then there is no harm in that.
Allaah knows best.