I don't know why Spock would say it that way.
Because it was the second pilot, the first time his dual ancestry was mentioned, and they hadn't yet settled on the specifics. Note that in "Mudd's Women," which was written around the same time as "Where No Man" and was a candidate for the second pilot, Spock is identified as only "part-Vulcanian." It wasn't until the series proper that they decided Spock would be fully half-human; in "The Corbomite Maneuver," they had Spock specify that his mother considered herself a fortunate Earthwoman, and in in "The Enemy Within," Spock referred to having "a human half... as well as an alien half." (His species name isn't mentioned again until Chapel calls him a Vulcan in "The Naked Time," though his planet was identified as Vulcan an episode earlier in "The Man Trap."
Anyway, since the "one of my ancestors" line was in a mutually teasing exchange with Kirk over a chess game, we can assume in-universe that Spock didn't intend it to be taken seriously or literally.
(The fact that Harry Mudd recognizes Spock as "part-Vulcanian" on sight, combined with Spock saying the Scary Balok Puppet reminds him of his father, suggests that full Vulcans were originally presumed to look more alien than Spock. I wonder what they would've looked like if they'd stuck with that idea. Probably bigger ears and eyebrows, greener skin, that sort of thing.)