Ichigo is the last remaining mew in his jungle so he decides to turn Shinji, a Team Rocket grunt, into a mew to help increase the population.
Zangetsu is a Mewtwo born of Ichigo’s and Yveltal DNA, thus making him technically Ichigo’s child, who destroys the lab they were born in immediately after gaining consciousness and seeks out his father in hopes of being accepted as his offspring.

To explain Ichigo’s coloration:
First generation Mews, also known as origin Mews, were pink in color. The second generation saw orange, purple, and blue hues as well as speckled and striped patterns popping out. As time went on and generations passed more colors and pattern variations became present, especially as more mews bred with other pokémon.