Do you know a proverb 夫婦喧嘩は犬も食わない? This も is "too/also" (or more accurately, "not~either"), and the nuance "even (if)" is given by the context (showing an extreme example), as same as your sentence.
Incidentally, indeed でも is originally made of the -te form of copula and も, but it's treated as a particle as a set. The meaning "even if you are ~" is from the context, not from で.