That's not nearly as common as you might think--among other things, it's a good way to ruin a several thousand dollar fursuit. (The best analogy I've heard is "Yes, there are furries who have sex in fursuits, but probably not more than people who have sex in space suits."[1])
The furry fandom is much less inhibited than general society--members tend to be willing to discuss sexual topics in a way that most people aren't. (This probably also accounts for the higher incidence of homosexuality[2], since it's not felt to be something that needs to be hidden.) However, the vast majority of furry-related sexual expression (as opposed to two people who happen to be furries having otherwise non-furry sexual relations) comes in the form of various types of media--published stories and images, but also private roleplay. In this regard, pet names based on animals are absolutely a part of "furry sex", certainly more so than dressing in animal suits during the act.
The furry fandom is much less inhibited than general society--members tend to be willing to discuss sexual topics in a way that most people aren't. (This probably also accounts for the higher incidence of homosexuality[2], since it's not felt to be something that needs to be hidden.) However, the vast majority of furry-related sexual expression (as opposed to two people who happen to be furries having otherwise non-furry sexual relations) comes in the form of various types of media--published stories and images, but also private roleplay. In this regard, pet names based on animals are absolutely a part of "furry sex", certainly more so than dressing in animal suits during the act.
1: http://www.adjectivespecies.com/2015/08/29/yiff-murr-sex-in-...
2: https://doi.org/10.1163/156853008X323376; poorly-formatted PDF at https://archive.org/details/pdfy-vF2uQGRhrwdL1oe3. "...they are less likely to be heterosexual compared to other males in the study."