No, sex is still a social construct in that it's a label made up in order to explain certain biological traits. It is also not a binary, but a bimodal distribution which is how most scientists treat it nowadays.
So yeah, it is assigned because nobody is looking deep into chromosomes etc at birth for the most part, they're merely looking at genitals and either assigning a sex based on that or in the case of intersex people often non-consensually 'correcting' them.
This is amazing! Thanks!