There are no exact synonyms

There are no exact synonyms.

That may sound a little extreme, especially given that thesauruses exist.

There is no pair of words where one can replace the other in all circumstances.

'Bail out the canoe with a bucket'

Can you replace 'bucket' with 'pail'. Of course. But can you say 'kick the pail'? No, of course not, that would be wrong. You can't always replace a word with its purported synonym.

Well, OK, there are some circumstances where there are exact synonyms. In technical circles, especially the sciences and math, there is a special way of attaching a word to a definition. In technical areas one 'stipulates' a definition of a word. That is, you give a word a definition that is simply a shorthand for replacing a word with its definition. You're stating authoritatively that for a word, you must treat it like an exact replacement. Often these technical terms are supposed to be evocative or metaphorical, supposed to give you a good idea of the intended meaning. But you can have whatever mental connotations that help you remember the true meaning but the true meaning is what has been stipulated, it doesn't matter, A=B and that's all there is no more no less.

But with non-technical words, there is no stipulation. A word is just a trigger for some associations. And if it sounds different, then there is no way it can be identical in all situations. Different stimuli can give different responses.

I will even go so far as to say that even a given word is often not its own synonym because all words have multiple meanings.  I'm not even talking about homophones (words that are spelled differently, but sound the same, like 'horse' and 'hoarse') or the other side homographs (words that are spelled the same but can have different pronunciations and meanings, like 'bow' a knot in a ribbon or tie, and 'bow' the front of a ship). I mean a word that is spelled and pronounced the same but has a different but related meaning. For example, 'run' is a verb to move fast by your legs, but is also a noun for a  long rip in a stocking or a small stream.

The point is that if you desire a synonym, you can get that from a thesaurus, but it may not slot in perfectly as a replacement. And even a single word may have many associations and alternate meanings that it is not good in that slot itself.