For parts of the human body such as the appendix, the term vestigial is often used.
Here is a definition: Used to describe something, especially a part of the body, that has not developed completely, or has stopped being used and has almost disappeared
Note that the definition does not exclude its usage for things that are not parts of the body.
Here are a couple of examples of its non-medical usage which definitely carry the meaning that they no longer have a purpose: As it turns out, these vestigial urban remnants that serve absolutely no purpose have a name.
They’re called “Thomassons” - 6sqft ..
a doorway like an ancient castle narrowing to a sort of stone tunnel as if for defense — one of those vestigial architectural features which Lanny had explained to Leutnant Rörich at the Château de Belcour Presidential Agent
The first article also proposes the term Thomasson for architectural features that serve no purpose yet are still maintained.
The term “Thomasson” has its roots in Gary Thomasson, an American baseball player who was traded to Tokyo’s Yomiuri Giants.
Thomasson was paid a fortune for a two-year contract, but eventually lost his luster and was benched for most his contract.
For Akasegawa, Thomasson was both “useless” and “maintained.”
ell.stackexchange.com/users/30978/JavaLatte
At first, I saw no problem calling it a vestigial ring , or a vestige an ornamental ring , or an ornament a remnant .
However, after searching around, I found this:
According to Brooklyn Brainery , who asked themselves the same question, those tiny handles are remnants from a time when maple syrup jars came in large five-pound earthenware containers, when the handles clearly served essential functional roles.
But over the years, due to the fact that we associate that container shape with maple syrup, the handles have been retained as a design element in the shrunken version we now buy, even though they don't serve any real function.
In fact, maple syrup handles are a classic example of a skeuomorph , a "retained but no longer functional stylistic feature."
( Source. )
From Wikipedia,
A skeuomorph is a derivative object that retains ornamental design cues from structures that were necessary in the original.
Skeuomorph is not a widely known word.
A quick search for some idea of the comparison gives the following.
So although skeuomorph has the definition we seek (and apparently it was precisely the word you were looking for), your audience is more likely to understand what you mean if you call the ring a vestige , or vestigial .
They'll get the idea that the ring is left over from a previous design or purpose, but that it is no longer necessary.
ell.stackexchange.com/users/36187/Em.
ornamental (adjective): serving or intended as an ornament; decorative. "Ornamental" conveys the basic idea that the handle is not functional, and would be appropriate for all audiences.
vestigial (adjective): forming a very small remnant of something that was once much larger or more noticeable.
"Vestigial" conveys that the ornamental handle was functional in an earlier incarnation of the object.
This term would be appropriate for an educated audience.
skeuomorph (adjective): an ornament or design on an object copied from a form of the object when made from another material or by other techniques "Skeuomorph" is a delightfully precise term, indicating that the vestigial feature is an intentional design cue.
However, since its introduction circa 1890, it has not been widely used by laypersons.
As such, this jargon is best employed when speaking to an audience specializing in design, marketing, engineering or a related field.
Also, most definitions of skeuomorph require the traditional object to be made from another material rather than simply a larger scale.
That said, this is still a very young word and could benefit from expanding usage.
ell.stackexchange.com/users/20377/james turner
As @oerkelens noted in his comment on the question, obsolete may be a useful descriptor, but it often refers to a whole object that is outdated and may have a newer version available.
I also thought of deprecated , but I think this word is primarily used in programming to refer to properties or algorithms that should not be used because newer versions exist that should be used instead.
ell.stackexchange.com/users/31774/Jed Schaaf
Speaking more generally - superfluous is a good word to describe something extra but unnecessary
ell.stackexchange.com/users/39999/web_bod
As an example the small ring in the maple syrup bottles: now too small to be usefull but still present
What is the word for a property no more necessary but still present in a object?
ell.stackexchange.com
<word-request>
ell.stackexchange.com/questions/99996
ell.stackexchange.com/users/39958/maborg
Vestigial conveys that the ornamental handle was functional in an earlier incarnation of the object. A  skeuomorph   is a derivative object that retains ornamental design cues from structures that were necessary in the original. However, vestigial is likely to be more widely understood. Ornamental or superfluous could also be used in this context. The word "skeuomorph" is the exact word to describe a derivative object with design elements that were necessary in the original, which were copied to the new form although they no longer have a practical purpose. However, since it is not widely used, perhaps "vestigial" would be more clear to the target audience. "Vestigial" is often used for the human body, but in its broader sense it means a small remnant of something much larger in the past, so it would work for the ornamental handle that was essential in an earlier form of the object. "Ornamental" could also work, since it is a widely used word that describes something decorative and conveys the idea that the element it refers to is not functional. Finally, in more general terms, "superfluous" also signifies something additional but unnecessary.
"Vestigial" is often used for the human body, but in its broader sense it means a small remnant of something much larger in the past, so it would work for the ornamental handle that was essential in an earlier form of the object.  The word "skeuomorph" is the exact word to describe a derivative object with design elements that were necessary in the original, which were copied to the new form although they no longer have a practical purpose. However, since it is not widely used, perhaps "vestigial" would be more clear to the target audience. "Ornamental" is a widely used word that describes something decorative and conveys the idea that the element it refers to is not functional. In more general terms, "superfluous" also signifies something additional but unnecessary.