There's nothing wrong with this, exactly. Phil could well be 1/64th Cherokee by blood. More likely, someone told him his great-great-great-great grandmother was Cherokee and he swallowed the story uncritically. Which is how it goes about 63/64ths of the time.
But the show has tried this gimmick twice, which makes the writers seem a little thoughtless and unoriginal. Not to mention desperate. Have they been listening to the critics who have pointed out how white the show is? Except for Gloria, who's a fiery Latina stereotype?
What better way to inject some color--literally and figuratively--into their white-bread characters? Many people claim to have Cherokee ancestors who are difficult if not impossible to verify. It's a cheap way to imply a character isn't as white and privileged as he seems.
If Phil is 1/64th Cherokee, you're supposed to feel 1/64th more sympathy for him, or something. The show can then proceed without confronting a single issue of race or culture. It's the TV equivalent of checking a box on a job application. "For the sake of the diversity police, we checked the ethnicity box for Phil this week. Done!"
For more on Modern Family, see History Class in Modern Family.