There are relatively few cases of true synonyms in English (or any language). There are subtle differences in meaning, register, etc that are recognized by native speakers.
I don't what you mean by a "true" synonym, but that is false. There are historical reasons there is a lot of word-doubling. The fact that synonyms might carry additional subtle connotations -- i.e. maybe you find "autumn" more poetic than "fall" -- doesn't change the fact that they are synonyms.
One of the main points of language is to evoke ideas in others' minds. "Autumn" and "fall" will normally evoke different ideas in US english (the former bringing about views of the cozier parts of the season, and the latter being more sterile, used to refer to a particular region of time). Maybe we disagree about what a "true synonym" is, but that distinction seems important to me.
Then by your logic no words can be true synonyms, because sounds themselves inherently evoke meaning [1]. And then the concept itself becomes pointless.
So I'd prefer to stick to concepts that actually do mean something, where "synonym" is two words that have the same primary functional meaning. Splitting hairs over whether autumn is "cozier" than fall does not change the fact that they are synonyms by any reasonable definition, or change the point that English has a hugely larger number of synonyms than any other language, for certain historical reasons.
