Mew didn’t give birth… anymore
Out of universe, the text you quote comes from a time before in-game breeding, Pokémon eggs, or anything of the sort. And while the same text from the first gen games about "giving birth" can be found in FRLG (where there is breeding), it doesn’t appear anywhere else. In LGPE, "We obtained a new Pokémon from Mew" while in the Pokémon Origins anime it’s "From the potential within Mew, we have managed to create an entirely new Pokémon: Mewtwo." To the best of my knowledge, Mew was the only time that a Pokémon was said to have given birth, but now it’s retconned away.
Other methods of creation
There are plenty of examples of Pokémon that are said by their Pokédex entries to be created through methods other than eggs, though many of them also hatch from eggs. Off the top of my head:
- Porygon (man-made)
- Many ghosts (such as Banette, originally a doll)
- Exeggcute (heads can fall off Exeggutor and form a group)
Many legendaries were said to have been created. For example, in the Sinnoh myths, Arceus (who did hatch from an egg) created the lake trio, and Dialga and Palkia:
Having tumbled from the vortex, the Egg gave rise to the Original One.
From itself, two beings the Original One did make.
Time started to spin. Space began to expand.
From itself again, three living things the Original One did make.