What does to enter a relationship with somebody mean?
"Pia Mia was introduced to Chris Brown's manager and record executive Abou 'Bu' Thaim in 2013. She began working on material with producer Nic Nac and has since entered a relationship with him, including a cover of Drake's 2013 single "Hold On, We're Going Home". "

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A relationship refers to a romantic relationship. If it is a different type of relationship it should be written with the other word to describe it: This sentence reads as though it should be "business relationship".

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Although it could mean to start a relationship of any kind I think it would be mostly used when referring to a business relationship as it's quite formal sounding language.

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Although "to enter a relationship with someone" can mean to date as a couple, I don't know if that's what it means here. Here, it might just mean that they are working together but, to me at least, it seems like an awkward way to phrase it.