Radius is the only word that means radius. Is there a word for that state of affairs? (you know, in the same sort of way that a plant genus containing only one species is called monotypic)
Half-diameter means radius. 
Yes, I know that many technical definitions don’t have true synonyms. Nor do some of the commonest words. What is the synonym of “the” or “you”?
Does that mean that the concepts have only one name? “plant genus containing only one species” is the same concept as “monotypic”. You can always express a concept in a variety of terms, even if you stick to a single language.
Of course, you can always coin a word to have that technical meaning. Mononamic, anyone?
For “the”, “ye” (though some would say that is just an alternate spelling of “the”). Some people might adopt a foreign article such as “le”, “la”, or “el” as an affectation, e.g. “I need to handle los action items.”
For “you”, “thou/thee”. In former times, these were the familiar forms (i.e. thou is to tu as you (back then, ye) is to vous), but nowadays they are interchangeable, though using thou/thee today is an affectation or a sign that you are reading an old document.
That’s a description of something, not the name of something.
But a simpler way to frame the question: is there a word to describe words which have no synonyms?
I mean, OK. ‘Radius’ is also a description, but not every description is a name.
“Ye” when used as a definite article – “Ye Olde Shoppe” – is just an alternative spelling for “the.”
“Ye” when used as the archaic subjective-case plural second-person personal pronoun – “Ye be strong, my friends” – is not just an alternative spelling for “the.”
This seems like way too common a situation to have a word describing it. I just looked around the room for objects, and the very first one I thought about was “door”. I don’t mean the passageway itself, but rather the physical, rectangular surface that hangs on the hinges, like this one. What else could you possibly call that object? “Bob, could you remove the _____ from the hinges so we can get this couch in there?”
I also see a banana, chalk (for the blackboard), and a bottle filled with a liquid called water. Am I missing the OP’s intent?
not really, but I don’t see why being common would deprive the phenomenon of a name. Synonymy is very common, and that has a name.
Semidiameter is an actual word meaning radius.
ISTM that every word has connotations and context, so that no word means exactly the same as any other – and since so many words have more than one meaning, few words always mean the same thing they do themselves. It’s not like we can say that the OP’s class of concepts are “unique reference concepts”: the word unique is steadily losing its grip.
But heck, let’s swipe a word from German - or make one up - it’s what we do in English, and German still sounds a bit tekkie. I propose “festnamen”. (Many years since highschool - the word makes sense? )
Door <- Hatch, Lid, Cover, Gate
Banana <- plantain
Chalk <- Dry-marker (depending on the actual board)
Water <- DHMO, dihydrogenmonoxide
Someone else can probably do better than ‘dry-marker’ for ‘chalk’, but I admit that I can’t, for the moment.
Fair enough. If you do find a word for it, it’ll likely be autological.
(etc.)
Perhaps you’re just joshing, but of course those words do not refer to the thing in the picture I linked to. (Same with your other examples…)
I don’t think this will be true. I’ve made a very similar argument about analogies in the past, but the analogies specifically require a difference of some kind. Synonyms don’t require such a difference, and often exist just because humans like coining words.
I reckon we could find examples of synonyms that have no significant differences in meaning and are just different words for exactly the same thing.
Possibly Sui generis?
I’d like it to be “mononymic.” Unfortunately that’s used basically as a synonym of “monomial,” having a one-word name (or, in mathematics, having one term).
But they are conceptually the same thing–the surface which occupies and closes off a portal.
The problem is, meaning is often derived from personal experience and connotations of words. At least one person is bound to assign slightly different meaning to certain synonyms. And whatever synonyms you can come up with, it’s possible to arbitrarily define slightly different meanings and connotations to them. After all, words are just signs that more or less arbitrarily ‘mean’ something, or refer to something. And there’s no definitive list either. If you discover a concept that only has one name to describe it, I could always come up with another one for it.
OK, but none of that matters to the question of whether we can have a word that describes a term with no synonyms.
It doesn’t matter that the real world is messy or fuzzy - it most often is - that has never impeded the invention of terms that assume a discretely-partitioned world.
Unsynonymed?
Synonymless?
Festname would be fixed name, not unique name. Namen would be names, plural.
Unique names would be “einzigartige Namen”, but you are probably looking for one of those bandworm words?
Unless you count spelling variations, such cases are exceedingly rare. The only one in English I can think of are “a” and “an”, which are two different words (insofar as they cannot be used interchangeably) with the same meaning.