No, it appears to be a different species, although it may be one obtained from people. Chlamydia pneumoniae (that affects koalas, and through my quick skimming, humans), is different from Chlamydia trachomatis (the human STD).
Many animals get Chlamydia (or Chlamydiophila), and it is not sexually transmitted like it is in people. Birds and cats are other common domestic animals affected by that species. Psittacosis, which can cause respiratory disease in people, is an avian respiratory disease caused by Chlamydiophila psittaci.
This is one thing that people confuse. Just because what causes disease in humans is contracted by sex, does not mean another species in the same genus can cause disease in another way in another animal.
All animals have some sort of herpes, for example… Most of them (or the most serious of them) are not STD. They can get some types of Chlamydia(phila)… I think they can also get a relative of Neisseria (gonorrhea) without intercourse.