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At every shop i've worked at: dealership, corporate aftermarket, or independent our technicians had a dress code. Usually a matching shirt/pants combination. The technican's clothing will make the first impression on your customer, and it will more than likely be the only impression, because he will not be talking to the customer.
Anything that is not part of the dress code set up for the shop is inappropriate. Most facilites do not allow hats, earrings, necklaces outside of workshirt, tanktops, shorts, ect.
It does not matter how nice the clothes are (could be t-shirts with shop's logo), but it is most important to an observing customer that all the technician's clothing match. Don't let one technician wear his undershirt while the others are dressed in uniform.
Make sure that everyone has their shirts tucked in (including belts). This provides a more professional look, and will be noticed by the customers. If the customer feels that the technican is professional looking, they will have more confidence in his abilities, and are less likely to look over his shoulder the entire time.
Please let me know if you have any more questions. I have worked in both the parts and service industries for multiple companies.