The closest I could find is non-random mating without resorting to words like "discriminating", "selective":
Even without the intervention of
farmers, most animals select mates
carefully--they do not mate randomly.
Charles Darwin noted this fact in his
1871 book Descent of Man and Selection
in Relation to Sex. He suggested that
mate selection is a powerful force of
evolution similar in its effect to
natural selection. This idea was
widely rejected in Darwin's time, but
later research showed that he was
correct.
see also assortative mating or assortative pairing
Assortative mating (also called
assortative pairing) is the phenomenon
where a sexually reproducing organism
chooses to mate with individuals that
are similar (positive
assortative mating) or dissimilar
(negative assortative mating) to
itself in some specific manner. In
evolution, these two types of
assortative mating have the effect,
respectively, of increasing or
reducing the range of variation (trait
variance), when the assorting is cued
on heritable traits. Positive
assortative mating, therefore, results
in disruptive natural selection, and
negative assortative mating results in
stabilized natural selection.
using those terms you could probably use something like "assortative selection" or "positive assortative selection". If someone was picky in the opposite sense (i.e. going for more diversity) then it would be disassortive mating (or arguably, "pairing"):
Disassortative sexual selection is a
form of sexual selection in which one
sex chooses the other, in such a way
that the offspring benefits from the
diversity of the parental genotypes
You can also try homogamy (or "homogamous")
Homogamy is marriage between
individuals who are, in some
culturally important way, similar to
each other. Homogamy may be based on
socio-economic status, class, gender,
ethnicity, or religion. It is a
form of assortative mating.It can also
refer to the socialization customs of
a particular group; such that people
who are similar in religion, class,
gender, or culture tend to socialize
with one another.
or even heterogamy or "heterogamous":
In a sociology, heterogamy refers to a
marriage between two individuals that
differ in a certain criterion, and is
contrasted with homogamy for marriage
or union between partners that match
according to that criterion. For
example, ethnic heterogamy refers to
marriages involving individuals of
different ethnic groups. Age
heterogamy refers to marriages
involving partners of significantly
different ages. Heterogamy and
homogamy are also used to describe
marriage or union between people of
unlike and like sex (or gender)
respectively.
so to answer you question - that person might probably a non-random mate selector/an assortative selector/a disassortative selector/a homogamist/a heterogamist