Where do Philip and Elizabeth's names come from?

Out of universe, they were not, from this interview with showrunners June Thomas and Joe Weisberg:

Thomas: When you named the main characters Philip and Elizabeth, were you thinking of the British royal family?

Weisberg: There’s a whole list of things that I’m going to have to claim I did on purpose. That’s one of them. Another is Clark for Clark Kent. Then someone told me that Marthas were what the British called secretaries.

In Universe, no, they were not named after the royals.

The characters in The Americans did not choose their American names, nor did the KGB just make them up. The illegals (a term for an undercover agent operating inside another country by posing as a native and without diplomatic immunity) used what are called "dead doubles." A dead double is an identity lifted from an actual person with a birth certificate (and, in the US, possibly with a social security number) who died, usually as a child. The characters in The Americans explicitly explain to their daughter that their names were lifted from Americans who died. This occurs in Season 5, Episode 12 at the 32:05 mark:

Paige: Where did our name come from? Jennings?

Elizabeth: Before we got here, our people worked it out.

Phillip: What happens in these cases is you need a new name to fit in where you're going. So they look for the name of someone who died; they're not using it anymore. So that's where it comes from.

Paige: Phillip and Elizabeth Jennings are dead people?

Elizabeth: Well, Phillip Jennings is... was. And Elizabeth Korman... who married him.

Paige: Do you know anything about them? Who they were?

Phillip: When they were born. Where they're from, same as us.

The dialogue is a little misleading because the real Elizabeth Korman and the real Phillip Jennings never would have married. First, giving married illegals the identities of people who had been married would create a risk of discovery. It is one thing to meet a person with the same exact name as someone you know who died. It is quite another to meet a married couple with the exact same names (including maiden name) of a married couple you knew who died. It also invites looks into newspaper wedding announcements with photos, etc. Second, dead doubles were almost always people who died very young and were born within a couple of years of the agents who would use their names (so they would look the right age). The Soviets collected identity papers for dead doubles from Americans sympathetic to the Communist cause. It would take some years to vet the names and match them to people, so given their young age when they got them and assuming dead doubles of roughly the same age, the real Phillip and Elizabeth would have had to have died as children. So in universe, the names were coincidental based on the available identities of the right approximate age for the two agents.

At other points in the series we learn that the real Phillip Jennings was born in Pittsburgh, so that's where Misha/Phillip claimed to be from, and the real Elizabeth Korman was from Chicago, giving Nadezhda/Elizabeth her fake hometown.