Rather than being about actual ancestry, the names for these foot shapes come from their representations in those cultures' art in Antiquity, and just describe the shape. At least for Egyptian, Roman, and Greek, I can't speak for 4 and 5.

You can see in this French pop-science site, which opens,

Egyptian Feet, Greek Feet, or Roman Feet?

Some people have green eyes, others blue. Some are blonde, others are brunettes. Feet are the same... we don't all have the same ones!

It claims that 63% of people have "Egyptian" feet, 31% have "Greek," and only 6% have square or "Roman" feet (I've also seen "Polynesian" in English). This English site adds "Peasant" and "Simian" foot shapes; they too are just using "Egyptian" and "Greek" to refer to the shape, not to ancestry.

The rationale, for Greek/Egyptian/Square, is that the "Egyptian" foot was artistic ideal in Egyptian art and statuary, whereas Greek art and sculpture, and subsequently Renaissance and classical-inspired art displayed the ideal "Greek" feet, but it's possible to find counterexamples, too. (Like, here's an Egyptian statue with "Greek" feet.) Whether or not any of this was ever based on an actual genetic/population difference between those two originally, I don't know. And I don't know where your image got Celtic and Germanic feet from.

Geneticist here.

There is some heritable variation in whether your second toe is longer than your first toe ("Morton's toe"). In fact it's a simple Mendelian dominant trait.

OMIM cites some ethnic differences in the frequency of this trait, thought not for these five (rather closely related) groups. I highly doubt these figures are based on any real evidence.

However, Wikipedia also cites some cultural associations, which I'd guess are the real culprit:

In statuary and shoe fitting, a more-protruberant second toe has been called the Greek foot (as opposed to the Egyptian foot, where the great toe is longer). It was an idealized form in Greek sculpture, and this persisted as an aesthetic standard through Roman and Renaissance periods and later (the Statue of Liberty has toes of this proportion). There are also associations found within Celtic groups. The French call it commonly pied grec (just as the Italians call it piede greco), but sometimes pied ancestral or pied de Néanderthal.[5]

I'll let the historians elaborate on that.