It struck me that a language could have a system of lexical associations that are somewhat gender-like without quite being a gender system. Let's begin with a basic, fairly gender-like version.
The ways in which this system expresses itself is in which nouns are used to express relations:
If a thing is the origin of something, it's either its mother or father; if it originates with another thing, it is its son or daughter; it goes together with something - brother or sister. Closely associated with something? Husband or wife. Other typically sex-specific terms are also associated, and human-metaphors are either fully male-centric or fully female-centric for any given noun.
However, we can also imagine other ways of expressing relations. Maybe the categories are streams and trees!
Origin - source, root.
'Offspring' - (part of) delta, branch (or nut, or seed, or fruit)
Associate - a different, named river that is culturally seen as close, a specific different species of tree
Intimate associate - a named tributary, the word for a major branch
Decline - drought, fall over
Increase - flood, grow