From Halopedia, the Halo wiki
The phoenix-like design of the bird is one of a number of allusions to Phoenixes within Halo Wars and its associated iconography. These references include the UNSC Spirit of Fire's logo and its ship class, the Phoenix-class colony ship. The references are a nod to Halo Wars' origins as an original IP codenamed Phoenix, prior to the project becoming Halo Wars.
List of appearances