There Are No Synonyms

This claim is not exactly true.
A synonym, according to our good friend Google, is defined as "a word or phrase that means exactly or nearly the same as another word or phrase in the same language," and there are certainly words that mean nearly the same thing as other words. 'Substitute', 'alternative', 'equivalent', and 'euphemism' are all listed as synonyms for the word 'synonym', for instance, and it is true that they all mean reasonably similar things.
Unfortunately, this is not actually how synonyms are taught to us. Somewhere along the line, we all had an elementary school teacher who defined a synonym as a word that meant the same thing as another word. Not a word that meant something similar. Not a word with a comparable meaning. A word that meant exactly the same thing as another word.

The same twitwho tried to tell us that commas go where you pause in speech, that we would never need to use cursive writing in real life, and that the stupid 'order of operations' is a reasonable way to approach math. Who even hired her?

The reality of the matter is that there is no such thing. The beauty (or infernal senselessness, depending on your perspective) of the English language is that every word has a wealth of subtle nuance that sets it apart from any other. To emphasize this a little, just by clicking on supposed synonyms I was able to get from 'synonym' to 'binge drinking' in a mere seven steps.
Seriously.
The reason for this is that English as we know it is the slow-witted, two-headed, inbred bastard child of Old English, French, German, Greek, Latin, and most likely a slough of other linguistic influences of which I am unaware. Therefore for every concept which we recognize, there is potentially a word for it that derives from every parent language. What's more, each one comes with its own cultural baggage that subtly affects its meaning.
Consider the example of livestock. While we raise them, we use the Anglo-Saxon-derived words 'sheep', 'swine', 'cow', and 'calf'. While we eat them, however, we refer to them with the much more Latin 'mutton', 'pork', 'beef', and 'veal'. Why is this? One possible explanation is that in early England, when the average community was a smallish farming community, these concepts referred to all the furry things running around outside of town, while in the notoriously luxurious urban sprawl of Rome, they would have been more familiar as the tasty things found on dinner tables.
My favorite example, that of 'friend' and 'companion', is very similar. The old English word 'geferan', which evolved into 'friend', is derived from the verb 'feran', meaning 'to travel'. In their warrior culture, your friends were quite literally the people with whom you adventured. Its supposed synonym 'companion', on the other hand, comes from the Latin 'com-', meaning 'together', and 'panis', meaning 'bread'; for the French who melded the two, your friends were the ones with whom you shared dinner.
Even today this distinction is subtly present. When you want to go out and do something silly, fun, and/or dangerous, you call your friends, while calling someone your companion has a much more formal vibe.

Though admittedly there were nine companions who proved a bit of an exception.

The point of this is that we, as a society, need to stop thumbing through our thesauruses looking for words that merely mean the same thing. Every single smallest word in our language has a subtly unique personality and identity, which is backed up by a rich cultural heritage all its own, and there's a beauty in that which deserves appreciating.