Sources note: Choosing only those embryos of species that fit the Darwin/Haeckel frame for teaching purposes – as opposed to a range of accurate depictions – isn’t the biggest problem, nor is exaggerating the similarities midway through development. Haeckel’s most serious misrepresentation is that he left out the earliest stages in embryo development – when various classes differ markedly.
Why would he do that? In order to demonstrate common ancestry through embryos, what you need is for them to all start out very similar and gradually diverge as they develop. And that does not happen. Of course, common ancestry can be true even if embryos do not demonstrate it. But if we believe there is sufficient evidence for common ancestry, why choose fake evidence to demonstrate it?
See Jonathan Wells, “Haeckel’s embryos: Setting the record straight,” The American Biology Teacher (May 1, 1999):
Differences among the four classes are evident even in the fertilized eggs: zebrafish and frog eggs are approximately the same size (about a millimeter in diameter); the chick embryo is a disk 3 to 4 millimeters in diameter which sits on top of a large yolk; while the human embryo is only about 0.05 millimeters in diameter (Figure 3, top row). The earliest cell divisions in zebrafish, frog and chick embryos are similar except for the fact that they are unable to penetrate the yolk in fish and bird eggs; but the earliest cell divisions in humans (and all other mammals) are completely different from the other three, since one of the second cleavage planes is rotated 90̊ relative to the other (Figure 3, second row).