The New Testament explicitly prohibits polyandry:
Thus a married woman is bound by the law to her husband as long as he
lives; but if her husband dies, she is discharged from the law
concerning the husband. Accordingly, she will be called an adulteress
if she lives with another man while her husband is alive. But if her
husband dies, she is free from that law, and if she marries another
man, she is not an adulteress. (Romans 7:2–3)
The Old Testament does not have such a clear prohibition, but the laws against adultery are usually interpreted to mean that polyandry is prohibited. For example:
If a man commits adultery with another man’s wife—with the wife of his
neighbor—both the adulterer and the adulteress are to be put to death.
Practically speaking, there are no examples of polyandry in the Bible, whereas there are many examples of polygyny.
As for the broader question, it would be a stretch to argue that marriage is gender-neutral in the Bible.
However, there is a general arc starting with male/female equality in the creation of humans in the first creation story (Genesis 1:26–27), to increasing inequality from the second creation story in Genesis 2 in which woman is created out of man (Genesis 2:21–23), to the punishment of Eve in Genesis 3:16 in which she is made subject to man, to the first example of polygyny in in Genesis 4:19–24, after which women are considered subject to men for the rest of the Old Testament narrative, and polygyny is common. But in the New Testament there is a move back toward monogamy, and a general increase in the status of women—without, however, explicitly re-establishing full equality between the sexes or universally prohibiting polygyny.
Still, the trend was set. Within a few centuries Christianity adopted a strict monogamous position, and the position of women in relation to men generally improved over the centuries. The same developments have taken place in post-Biblical Judaism as well.
For an analysis of the first three chapters of Genesis from the perspective of gender equality or lack thereof, see my articles, Man, Woman, and the Two Creation Stories of Genesis and What are the Roles of Men and Women toward Each Other and in Society?