Wellllll....technically he is right, it is a philosophy....because it is non-theist....it does not believe in a Supreme Being, or some Supernatural Entity.
But in fact, Buddhism still contains many of the characteristics of a system of spiritual practice and belief, yes.
You might say it's a very UN-ordinary form of religion.
Taoism is based heavily in philosophy as well as spirituality, and Buddhism is primarily based on self-evident truths. Very different from the Judeo-Christian idea of religion.
However, both have supernatural aspects (reincarnation, spirits, etc), and therefore are religions.
That wouldn't be true for either Buddhism or Taoism. Although both are deeply based on their respective philosophies, both also have a religious aspect.
Lord Buddha gave in a set of sermons that tells and teaches us how to live a right life. So, it is always considered to be a way of life. Yet, no harm in branding it as a religion called Buddhism. Basically, religion needs scriptures. Buddha's sermons and sayings form the scriptures of Buddhism.
Yes it is. Now a day anything people follow is a religion. But, the big difference between those religions and Christianity is, Christianity is based on a relationship with Christ. There are commandment and laws, church stuff, but it comes down to one thing, Do you have a relationship with Jesus Christ?
Buddhism is definitely a religion. Some people claim that it is an atheistic religion and that Buddhists are technically atheists, however I would say that this is a rather twisted and not very useful definition of the term 'atheist'. In any case this may be what's confusing you (or your teacher, for that matter).