I'm referring to a construction in a text where two options are provided, of which only one will be valid in practice. For example:
The user has the option to choose his/her own avatar.
In these cases, the slash is being used to express exclusive disjunction (one or the other, but not both).
I know of no conventional term for such phrases, but given the above, you might call them disjunctions, exclusive disjunctions or 'either/or' phrases.
You might call the slash itself a disjunctive slash or an exclusive disjunctive slash.
You are asking what people actually call this type of construction. Legal experts have referred to it as a “linguistic aberration” and a “disaster” because of the harm caused when trying to interpret contracts that use it. These terms, in turn, have been criticized as “somewhat disproportionate to the amount of harm it [and/or] causes”.
Outside of imprecations, there seems to be no accepted term for this construction.
Quotations are from Revisiting the Ambiguity of “And” and “Or” in Legal Drafting, Part VI, “And/Or”, by Kenneth A. Adams and Alan S. Kaye. [PDF]