Not saying there are the same character by the time we meet Nu-Khan. Yes, I see them as being the same man before the split in the timeline; the same historical figure and the same actions undertaken. I think STID's ideal that he's a war criminal and a genocidal supremacist , jives just fine with TOS and his smug superior attitude. And fits into the ideal that TNG+ introduced that another Khan must be avoided at all costs. But we see two different men when each Kirk runs into them. Nu-Khan was humbled, leashed, and had his crew (his family as his calls them) used against him. What we see is a man that's been used and looked down. And abused animal that's turning on it's master. In Space Seed we had a Khan that was woke up, idol worshipped, and had the senior staff (minus Spock) in awe of him. Khan woke up and saw a world that still admired him.