Are there examples of exact synonyms in English? That is where, at least in some specific context, you can you use either word and no one will say "one is old fashioned" or "one is dialect" or "they have slightly different meanings" or anything similar?
A particular language is not a computer language. Human languages are not designed, aren't made with deterministic parsers and semantic mappings (many programming languages aren't either!). Any attempt to give specific rules to human language is only an approximation to something that is somewhat fluid to begin with. Dictionaries didn't come first; the use of the words just is and lexicographers are just trying to capture meanings as well as they can.
As to vocabulary, people will use words that work for them, and sometimes adopt others they hear. Meanings are in peoples heads and we can only access those meanings through mapping an unshared (non-telepathic) cognitive situation to sound streams. This leads to the slightest drift of meaning around a sound sequence (word). (I'm not even discussing how the sounds change too.)
There are many ways to define 'define' or give the meaning of 'meaning'. They are all just attempts.
denotative meaning - a formulation of semantics in an unambiguous, non-vague, consistent and complete logical language.
connotative meaning - connections, collocations, contexts, vague feelings of nearby-ness.
stipulative definition - a meaning is assigned ('stipulated') for a given sound sequence
usage definition - a given sound sequence is used
Human language only follows 'usage' definitions. Mathematics, the sciences and technology do have a good portion of their vocabulary stipulated. Someone says "From now on, 'blurp' means 'take an allen wrench at 45 degrees from the plane but matching the hex hole and twist counter-clockwise'". Now 'blurp' has that stipulated meaning, and if someone uses 'blurp' for '30 degrees' or on phillips sockets, etc, then they are using the word 'blurp' wrong.
Take for example 'pail' and 'bucket'. What do they mean? They are pretty obviously refer to the same kinds of object, a cylindrical slightly conical receptacle open on one end.
Can two different sound sequences have the same denotative meaning? Yes. This is very easy in stipulative definitions. Also fairly easy (as in the case of pail and bucket) when there is a simple easily verifiable statement. But... sometimes things are left out. There may be subtle things left out of the denotative definition. And the connotations are almost always left out.
It is extremely difficult in usage definitions because there are so many different contexts that a word can appear in that probability wise it is unlikely.
Now to the question: are there any exact synonyms in English?
In technical English there certainly are (these might be called analytic a priori truths)... but you know, I just gave a speech above about stipulative definitions, but even there there is semantic drift. Even in math, the terminology changes because we're human and even technical terms lose their edge, get associated with some things rather than others.
But in non-technical language?
There are lots questions about two given words is X an exact synonym of Y. A lot of times the answer is, 'Wow it sure is hard to tell those two apart... sure, let's call them exact'. But the world is large and there are so many things we don't know about what we know (about how we use words). It's easy enough to reflect on one's inner connections about a word ('pail and bucket are identical! except I think of a pail as never being made out of wood')
So the answer really is
There are no exact synonyms.
- There is nothing here that is particular to English. This could easily be asked on Linguistics.SE or Philosophy.SE.
- Although there are two perfectly good possible duplicates, the title here is very specific and I can find no response elsewhere here on ELU that addresses this directly.