I'm very bad at naming stuff, so instead of trying to find good names... I find "good" reasons for the names being bad.
For a concrete example: I have a place named NML (Noon Moon Loom), which is an mouthful to actually say, and only slightly less so to say with initials. So I plan to, eventually, have a character note how horrible the name is, and another explaining that it came to be because people didn't want to refer to the place with its actual name (since it is a brothel), so they referred to it with other names: moon, loom, and others. During a rebranding the brothel decided to go with those instead, but as the management couldn't settle on one, they compromised by choosing some similar names each liked, and using them all. Now the building has neon signs that sequentially say either "Noon," "Moon," and "Loom," and eventually, as new people got to know the place, they started calling it NML.
Depends on the world they are set in, but in most cases I think the answer would be yes.
In a multi-species world, for example, even though furry species would have less incentive to turn to clothing solutions, both in the temperature and resistance departments, there would still be other species for which clothes would provide the same advantages it provides humans. Cold-blooded species are good candidates for clothes-as-temperature-control, and hairless species for clothes-as-defense-mechanism.
On single-species worlds, there's still, well, the world. Weather changes more rapidly than species adapt, so clothes could be invented as a way to deal with a suddenly colder climate.
If that doesn't fit, there's culture. Maybe, for centuries, taking care of their fur coat, and keeping it looking healthy and beautiful, has been an almost impossible endeavor. Clothes exist only to help the worst of the injured and diseased (for example, helping burn victims dealing with temperature management, or letting people infected with a hair-losing disease keep presentable), but the richer folks co-opt those clothes, decorate them, and start wearing them to hide the inevitable blemishes on they fur coats, and it becomes a status a symbol that eventually trickles down into lower classes. Many years later, once ~~skin~~ fur care starts improving, taking care of your fur becomes much more feasible, and the richest shift into wearing less clothing, it propagates, and now the only clothing that survives is that which either indicates status (displays of rare materials or colors), or has practical uses (pockets). Or maybe a culture of "nakedness" being seen as reprehensible has already developed, so while the richer folk are able to get away with it to some extent, the low/middle classes doesn't feel like they can, and they keep using full clothing, and it becomes a class discriminator.
I think that, for those cases, accessories would be a more obvious choice. Pendants, collars, hats, arm- and tail-bands, etc. Clothes as we know them, a full set of cloth from head to toe, covering everything, seems like would be a harder sell to come naturally without external factors driving them in. (Or I just lack imagination in this particular scenario.)
(And shoes! As much as we, furries, like our beans exposed, shoes are something that I think every tool-capable civilization would eventually create. We even created horseshoes for horses, so not even hooved species would be spared the inevitability of shoes.)
That's a very interesting case!
A world were different species have different sizes/needs regarding clothing is a world where clothing resists industrialization and mass production. Mass produced clothes could be derogatorily generic. Unless you were one of the lucky species/individual that fit the "standard" form that was profitable to produce, you would only buy those if you didn't have money for tailor-made clothes.
Each town would have their own tailors, who knew how to produce adequate clothing for the species in those towns. Before you moved into a town you'd have to check either tailors or seamstresses, to see if they were able to produce clothes that fit your species, else you'd have to choose another town. These clothing specializations could be a factor driving group/society/culture formation, and there could even be cultural conflicts between species that were usually catered for, and those who weren't. It would be a factor uniting certain species together, and keeping others segregated.