Hmmmm . . . I’d say ‘bird’ is roughly equivalent in level of casuality and contemporary with ‘bloke’. But while men may refer to each other as blokes (’Met this bloke in the pub’) I’m not sure women would refer to each other as birds in the same way. ‘Alright my bird’ is used in this part of the UK (South West) by women to each other but it’s an affectionate greeting rather than a reference. Most women would find being referred to as ‘bird’ - outside friendly jokey circumstances - as decidedly patronising.
So, can’t really think of anything equivalent at the moment, I’ll keep cogitating.