People can identify themselves however they see fit. Liberalism, in the classical sense, demands that the state protect this individual right. Thus its elevation of such principles as freedom of speech/expression, equality under the law, etc. That said, it does not demand that every government form, school teacher, or employer anticipate and accommodate every possible identity people might adopt.
This current obsession with identity undermines the very goals it claims to promote. The problem, ultimately, isn’t a lack of acceptance or tolerance, though there is certainly plenty of that out there. The problem is neither people nor their institutions have the mental/physical capacity to track an infinite variety of identities. It is simply unreasonable for a student to expect everyone to remember their chosen identity, let alone a professor with potentially hundreds of students. This barrier becomes even higher if we expect everyone to know what our chosen identity means to us. This problem becomes worse still, if that’s even possible, if we assume identity is so fluid from day-to-day that the pronoun someone used yesterday is potentially no longer valid today. Not everyone, trans or otherwise, is taking identity to such untenable extremes but some are.
To the extent the LGBT community (or any other group) insists that lapsing into the familiar sexual (as opposed to gender) binaries of he/she or his/her is a sign of bigotry or disrespect they only foster the bigotry and loathing they are supposedly trying to overcome. Most of us frankly don’t care how you identify, though we do care that everyone benefit from living in a society that champions universal human rights. So meet us halfway and pick one of the two or maybe three pronouns that we can easily remember and try to stick to it because people are never going to be able to keep track of much more than that no matter how hard they try or how well meaning they may be. Besides, if identity is really so fluid it can be anything at all, as some seem to think, it is meaningless so what difference does it make really? Sometimes society does have to make adjustments and should but there are limits. Sometimes people just have to go with society’s flow a bit too. In a truly civil society accommodation is, after all is said and done, a two way street.
Of course, in reality identity is fluid more like molasses is fluid and less like water. I am more than a little troubled by the implicit blank slate notion of human nature and its capacity to change from one thing to another from one day to the next some university professors seem to be advancing. How can an identity that is so changeable and fickle really mean anything to anyone? Described in this way identity sounds more like a curse to be avoided than something to be celebrated.