No. Dragons, which are reptiles, are in no way related to phoenixes, which are birds. Phoenixes are classified as 'pseudo-dragons', meaning false dragon. Pseudo-dragons are usually mistaken for dragons, as the phoenix might be, because it is a creature of fire. Other psuedo-dragons include basilisks, cockatrices, sea serpents, and komodo dragons.
AS Dragons was created in 1938.
Phoenix or dragons?
Imagine Dragons was created in 2008.
South Dragons was created in 2006.
Lucky Dragons was created in 1999.
Engadine Dragons was created in 1958.
Dragons of Deceit was created in 1985.
Dragons of Argonath was created in 1998.
Dayton Dragons was created in 1988.
The Dragons of Eden was created in 1977.
Pocket Dragons was created in 1989.