Are charisma and character synonyms?
Charisma means being full of character, so in some sentences yes, but that is entirely coincidental of your reasoning.
Is every synonym of a word's synonym also a synonym of the first word?
No. This is because not every word has the same meaning, and many words have multiple meanings. Even if a word has 15 different synonyms, not all of those words hold exactly the same meaning. However, in certain contexts, many words are interchangeable with one another.
Example: Anger and rage are synonyms, they have almost the same meaning... until anger is being used as a verb. "You anger me." makes sense but "You rage me." does not. Rage has a different forn for that function, "enrage".
Because words may have considerable overlap and contextual interchangeability, we categorize them as synonyms of each other for their individual similarities. Because not every word has exactly the same meaning, they don't perfectly overlap with the same group of words. While a synonym of a synonym may also be a direct synonym of the first word, it would likely already be listed as such in a thesaurus if it were.