From Janis Hutchinson comes this helpful explanation:

Why won't the mothers in the FLDS Church say which of the children are
theirs?  It may be because in many fundamentalist groups, ALL the women are
considered the mothers.  I know, that doesn't make sense.  But I know, first
hand, that in one fundamentalist group in St. George, Utah (The Church of Christ
Patriarchal led by John Bryant, of which I was a member for a short time), the
principle was this:

Since all the women under one roof were considered sister wives, all their
children were taught to call everyone of those wives . . .mother.  Therefore,
while each mother in the FLDS would naturally know which child she actually gave
birth to, perhaps they made some covenant to consider all the children of their
sister wives as their own, and by claiming any one child
as theirs would be considered selfish and a violation of the concept.  And if
the CHILDREN were asked who their individual mother is, they might honestly not
know.  This is the way the children were in the St. George group.  They really
didn't know who their natural mother was.  Please be aware that just because
this was the concept in the Utah group doesn't necessarily mean that this is
the principle that is being practiced in the FLDS Church.  But, most
fundamentalist groups do copy each other.
For more information, see The Mormon Mirage 3rd Edition: 
 A Former Member Looks at the Mormon Church Today 
(Zondervan, 2009)
Also available as an audiobook and as an
 expanded-text E-book for Nook, Kindle and other reading devices.