Maybe you have a more independent mindset having gone into a PhD program a little later than most, but the whole point of a PhD program (at least in the sciences) is that it’s an apprenticeship. You study under an established researcher using their grant, so it’s not “your” paper. You are supposed to work together using grant money from your advisor.
If you have obtained grant funding on your own and are working independently on a novel research project you thought of yourself, then you can call it your paper. But that scenario usually doesn’t happen, because it’s hard to come by funding without a good proposal, and it’s hard to write or qualify for a grant without the training one gets in a PhD program.
If you are working using grant money, lab equipment, lab space, data, models, software, or methods acquired and developed in your advisor’s lab, then even if you write an entire paper yourself it’s still both your names that go on the paper. I’ve had a few like that and was glad to share the credit, because it wouldn’t have been possible otherwise.
