How did a buddhism become a religion?

It would depend on how you define 'religion'.

I see religion as a means of preserving teachings in the form of verse (oral or written). This might incorporate rites and rituals as the teachings are passed down.

So, in answer to your question, I'd say it was after the second or at the third council (~220 BC) when distinctions between reverential, doctrinal and disciplinary practices started to appear. But this is when religion becomes sectarian.