Any shots can produce a discoloration of the skin or in this case the oral muscosa. The usual cause is nicking a small capillary, which then leaks a small amount of blood before it clots and is later repaired.
No, normally dental shots will not leave you with a bruise. They will be painful when the medication goes in, but your dentis will give you a topical numbing cream first. Your jaw may be slightly sore the rest of the day.