People who are born intersex are not biologically male or biologically female, correct.

If you wish to use standard terminology from the scientific study of sex determination, you may be able to designate someone as a (partially) masculinized female (given that the underlying regulation is female, but because of some abnormal regulation, at least some tissues have adopted a male fate) or a (partially) feminized male. However, this only is an option if you actually understand the what is causing the phenotype and the regulation is of this type (i.e. fundamental signaling is one type, but some abnormality in regulation leads to the signal being flipped in at least some tissues).

Otherwise, the question makes about as much sense as saying: what color is "purple", "blue" or "red"?

At the level of society, people may prefer to identify as a man or woman--taking a traditional binary gender role is a cultural phenomenon. When it comes to the biology, however, intersex people are intersex.

(Where exactly to draw the line between intersex and male or female with an unusual development of some sexually dimorphic characteristic is not entirely clear, but that's how dealing with messy reality is: you first label the most obvious things, then label the exceptions, then label the exceptions to the exceptions, etc..)