Philosophy makes a distinction between the identity of material objects* (and immaterial objects if those are thought to exist) and what is called "personal identity." From the Stanford Encyclopedia of Philosophy's article Personal Identity:
Personal identity deals with philosophical questions that arise about ourselves by virtue of our being people (or, as lawyers and philosophers like to say, persons). This contrasts with questions about ourselves that arise by virtue of our being living things, conscious beings, material objects, or the like.
In terms of the identity of a material object, philosophers would argue that a transgender person has the same identity as before. From the SEP's article on Identity:
A distinction is customarily drawn between qualitative and numerical identity or sameness. Things with qualitative identity share properties, so things can be more or less qualitatively identical.
A transgender person will have a massive amount of qualitative similarities with the person they were before coming out as transgender. These include:
Causal history of physical interactions
Genetic details and natal history
Collection of memories and historical subjective experiences.
The history of a person, the actions they've taken, who their parents are, the event of their birth, all of these and more are enough qualitative facts that are enough to say that someone has the same material identity as they did before. It is the same reason we would argue someone is the same person at one moment that they are one moment later.
In terms of personal identity, there are reasons to argue that a transgender person may be a different person than before once they come out. From the World Health Organization:
"Sex" refers to the biological and physiological characteristics that define men and women.
"Gender" refers to the socially constructed roles, behaviours, activities, and attributes that a given society considers appropriate for men and women.
So, taking the definition of gender to be a collection of rules, behaviors, activities, and attributes of an individual, we can argue that a change in these would fall in line with the definition of personal identity. Someone's personal identity can be seen as the answers to the question "who am I?" that they would give.
Consider that there is a transgender person named Alice, and Alice lived for twenty years as a man. At that point in her life she realized that she was transgender and wanted to be treated as the woman whom she felt she was. If we had asked her when she was 16 who she was, (maybe at that time it would have seemed appropriate to say "who he was" depending on the circumstances) she might have said something about her interests and her desired career and being someones son and so on. However, after she has had her revelation about her gender, her answers to that question would completely change. She would no longer identify herself as someones son, she would identify herself as a woman and a daughter. In this way, her personal identity has in fact changed with time. We could also consider the situation in which Alice as having known since she was 6 that she felt more like a girl and that was how she wanted to be identified. We would then need to look at her different behaviors and activities, as suggested by the WHO definition of gender, to evaluate whether or not she has had a change of personal identity.
At any rate, personal identity is different than the general metaphysical idea of the identity of a material object. It would not be well supported to argue that a transgender person is a different material object before and after they come out. In terms of personal identity, philosophy treats these issues by asking about things like social and psychological identification, as well as others. A transgender person may be, and most likely are, a different person after their transition, in terms of personal identity.
In regards to the three points made at the end of your question, a philosopher might make the arguments:
a) They would agree that they're the same material object, with the same qualitative properties of identity outlined above.
b) The metaphorical sense would be what was outlined above as personal identity, in some respects their personal identity has changed.
c) This depends on the person, if the person has felt since they were very little that they knew they were transgender then maybe that aspect of their personal identity hasn't changed, but there might be other aspects of their identity such as their actions, how they dress, how they talk, etc. that have changed after they came out.
*Additionally I'd like to say, of course I am not suggesting that we should treat people "like objects," I am just using a metaphysical distinction. Material objects are things that are created from matter and physically exist; the societal and cultural issues of referring or treating somebody as an "object" is a completely different matter. I'm not mixing the two.
