In short, "she/they" is the most common way for a person to indicate that they go by "she/her" or "they/them" pronouns, likely with a preference for the former. It is not incorrect.
Note: While "she/they" is commonly used by feminine-leaning non-binary people, a great deal of women (both cis and trans) also go by it.
There is no governing body for the English language, and the bodies that do try to govern it (e.g., APA, MLA, Chicago) do not have guidance on this because this just doesn't come up in formal writing. Therefore, we have to look at what usage is. In my experience, what's most common is:
- Subject/object when the person is using one set of pronouns. For example, "she/her". This is what the Trevor Project uses.
- Subject/subject when the person is using two different sets of pronouns, like "she/they". This has a nice parallelism to it. See also Washington and Lee University's article on the subject. (Note: The order may matter; usually the first one is what the person prefers to go by, though I also know some people for whom it is the opposite, usually listing a binary pronoun first but preferring "they", listed second.)
These two formats are what comes up for the vast majority of the results in searches for things like "pronoun pin", which brings up examples like the Human Rights Campaign.
However, there is variation. I've also seen the following, less commonly:
- Subject/object/possessive, for a single set of pronouns, such as "he/him/his". Sometimes it's the reflexive instead at the end, "he/him/himself", like in this Columbia University article. According to an NPR interview (no relation to me), this format used to be the most common, before the two pronoun form (e.g. "he/him") became more popular.
- More than two subject pronouns, to indicate more than two acceptable sets of pronouns. I just recently met someone going by "he/she/they", though it is uncommon.
- In the case of "it/its", the possessive is used, probably because "it/it" would be extremely weird. (Note: This pronoun set is pretty rare, but some do use it: Here’s why some LGBTQ youth are now embracing the nonbinary pronoun ‘it/its’)
- "Any" (meaning any pronoun), sometimes (very rarely!) paired with another pronoun which is usually itself in subject form. For example, I know someone who's going by "he/any", which means he prefers masculine pronouns but any pronoun works.
In the case of your suggestion, subject/object representing two different pronoun sets, I don't think I've ever come across this format before. However, it is evidently used albeit with little frequency; the Colorado government HR does list "he/them" and "she/them". I had to dig through Google to find this example.
Any of these options would at least be understood within English-speaking LGBTQ communities without question, and by a great deal of people outside them.
