I can guarantee that the kickers use the same balls as the offense and this has been the case for over 20 years. Each team provides their own footballs, and they can be different brands and vary (slightly) in length and circumference. They are weighed and measured and checked for proper inflation by the officials prior to each game, and marked by referee. There are generally six balls for each team (more if wet weather.) Kickers use whatever ball happens to be on the field when they kick.
In the NFL there are different balls for kickers.