Only the strongest of marketing campaigns has this kind of resonance. There are certainly other characters that have strong brand associations and have entered into iconic status -- Ronald McDonald, Toucan Sam, the Geico gecko are all pretty good examples -- and they have done so with heavy repetition, strong appeal to consumers and the ability to instantly convey the same values as the brand. On that level, the Energizer bunny has had enduring appeal that seems like it will never run out of steam.

An interesting tidbit about the Energizer Bunny, is that the character was created in response to Duracell letting their trademark lapse on the Duracell bunny, a character that it still uses in other parts of the world. The Energizer bunny was just a spoof on the Duracell character, but resonated with consumers to a much higher degree, creating an instant marketing hit. The bunny has been featured in the copany's marketing ever since, lending strength to its iconic status.

This represents the downside of the iconic bunny, in that it is more difficult for the company to do other things in its marketing -- Energizer is forever associated with the bunny, and with batteries, even though it is now trying to go into many other businesses. So while it is generally great to have an icon, there is some downside, like an actor typecast into one type of role.
References

Garretson, J. & Burton, S. (2005). The role of spokescharacters as advertisement and package cues in integrated marketing communications. Journal of Marketing. Vol. 69 (2005) 118-132.

Parekh, R. & Neff, J. (2013). Energize shifts marketing duties for several brands to JWT from Grey. Advertising Age. Retrieved April 19, 2014 from http://adage.com/article/agency-news/energizer-shifts-brands-jwt-grey/243735/