Certainly Buddhism has strong elements of both philosophy and psychology.
Buddhist philosophy Buddhist teachings do not require a suspension of the intellect by demanding a belief in scientifically implausible creation myths.
There are no 'revealed truths' (ie doctrines which come from out of the sky and must be believed on the basis of faith rather than reason). Buddha encouraged his students to test his teachings against their own reason and experience. Only by thoroughly challenging the teachings can one gain confidence in their truth.
One of Buddhism's main philosophical components is its ontology - the study of how things exist. A common misunderstanding is that Buddhists believe that 'things don't really exist' or that 'nothing exists'. In fact Buddhists believe that nothing exists by its own nature. All produced phenomena exist in dependence upon other phenomena - every cause is itself an effect of another cause. A table does not exist by virtue of it's innate 'tableness'. It exists due to the timber and the joiner, and its possessing a flat surface, a certain number of legs etc. It also exists by identification with the 'tableness' that is present in the minds of the observers (but not in the table itself!).
Tracing things further back, the timber exists in dependence upon acorns, soil, sun, rain etc, and the joiner exists in dependence upon his mother, father and the midwife.
In Buddhism, relationships such as cause and effect, structure and components, observer and observed are regarded as more fundamental aspects of existence than actual 'things'. Even the mind is not a thing or a substance. The technical Buddhist term for the mind is the 'Mental Continuum'. In western terminology we would regard Buddhism as a Process Philosophy.
Buddhist psychologyBuddhist psychology is intended to be used for improving our state of mind. It is an applied science and is not usually presented as an abstract or academic discipline, because in order to understand it Buddhists are supposed to 'walk their talk'. Practices include meditation, visualisation and mindfulness throughout the day.
When nineteenth century Europeans first studied Buddhism they were impressed by the rational aspects but were perplexed by some of the powerfully emotive and sometimes disturbing symbolism and visualisations. They ascribed these 'tantric' aspects to the corruption of a rationalistic philosophy by later mixing with primitive folklore and Shamanism.
Then along came Freud and Jung.
Buddha had recognised the importance of the subconscious activities of the mind, both individual and collective, 2400 years before the founders of Western psychology. He knew that purely rational arguments were insufficient to motivate a deep and lasting transformation of the mind. The practitioner also needs to harness and redirect the powerful emotional currents which well up from the depths. Jungian psychologists discovered that the vivid symbolism of tantric art and visualisation involved the use of 'archetypes' - ancient patterns and symbols in the human subconscious which can be invoked to produce powerful emotional responses.
So why is Buddhism regarded as a religion?The reason Buddhism is regarded as a religion rather than a form of humanism is that it is primarily concerned with the long term future of the mental continuum rather than with just this single limited lifetime. Buddhists do not believe that the mental continuum is dependent upon physical 'things' such as the body or brain for its existence. In fact many Buddhists would turn this view on its head and claim that the way that physical things exist is dependent upon the mental continuum of the observer.