I don’t think that’s true. For example you won’t find many Remainers claiming that the BBC was pro-Leave. Nor would you find many opposing Scottish independence saying the BBC was pro Scottish independence. It clearly does “take sides”.
The traditional left-right axis doesn’t work as a model here. The BBC’s bias is towards the opinions and interests of upper-middle-class Londoners, some of which would be traditionally be considered left and some of which would be right.
Likewise, if you look at something like The News Quiz, that and the presenters/guests are left wing and pro-Remain.
Those reporters and editors put out biased content from the left and right.
There are good arguments that the BBC do not criticise the current government and some people take that to mean that they support the government. That is exacerbated by the fact that David Cameron appointed prominent Conservatives to directorial roles in the BBC.[0]
But yes, the news they put out is not unbiased- it’s just that they have bias in all directions depending on who is writing and editing.
[0]: https://www.ft.com/content/7ba884c2-176d-11e6-b197-a4af20d55...
Their positions on most issues conform to that faction.
The reason I'd describe it this way is that it's much more apparent on social issues and certain aspects of international relations, where they don't even seem to understand that other viewpoints might exist. On economics they seem to have at least some awareness that there are a spectrum of viewpoints and take a more consumerist view as often as they do a soft left view (although they still never really seem to consider libertarian or harder left viewpoints).
IMO, the BBC can't be very biased because all sides accuse it of being biased against them!