Purely grammatically speaking, there are actually five inflectional forms that would need to be supplied: nominative (I), accusative (me), dependent genitive (my), independent genitive (mine), and reflexive (myself). I have also observed that people normally supply two (e.g. they/them). Note that this is indeed incomplete: what, for example, is the reflexive form? Is it themselves or themself? However, people sometimes do supply more than two. For example, according to this CNN report,
recently, Sen. Kamala Harris introduced herself at a CNN LGBTQ Town Hall by stating that her pronouns are “she, her and hers.”
I would speculate that the most common form of indicating the pronoun preference in writing,
(they/them)
(she/her)
etc.
i.e. two inflectional forms separated by a slash and in parentheses, has become a way to signal—a code of sorts—that these are indeed the person's preferred pronouns without the person having to write out the full sentence My preferred pronouns are… After all, giving just one may be a bit confusing as to what it is supposed to mean. (EDIT: this is point 2. in the answer to the question linked above.)
Discussion
Further down in the CNN report, it says,
The most common third-person singular pronouns are “she/her/hers” and “he/him/his.” “They/them” can also be used to refer to a single person, while some people use gender-neutral or gender-inclusive pronouns like “ze/hir” (pronounced zee/here) instead. Some people might not use pronouns at all and go only by a name.
Here is the 'traditional' list of first-person pronouns. The categories are NOMINATIVE, ACCUSATIVE, DEPENDENT GENITIVE, INDEPENDENT GENITIVE, and REFLEXIVE (CGEL, p. 327):
DEP IND
NOM ACC GEN GEN REF
I me my mine myself
he him his his himself
she her her hers herself
Here are five sample sentences, illustrating the use of these five inflectional forms:
NOM: I go to work.
ACC Words can't hurt me.
DEP GEN: That's my umbrella.
IND GEN: That umbrella is mine
REF: I may hurt myself.
Here are some gender-neutral third-person singular pronouns that have been proposed (source):
they them their theirs themself
(f)ae (f)aer (f)aer (f)aers (f)aerself
per per pers pers pserself
xe xem xyr xyrs xenself
ze/zie hir hir hirs hirself
zie zim zir zis zieself
sie sie hir hirs hirself
ey em eir eirs eirself
ve ver vis vis/vers verself
tey ter tem ters terself
e/ey em eir eirs emself/eirself