> "its own reference frame"

We don't use inertial frames of reference in General Relativity because in the presence of real gravity, there are (strictly speaking) none anywhere. [1]

There are however static spacetimes that admit inertial frames of reference. Flat spacetime aka Minkowski spacetime is an example. That is the spacetime of Special Relativity, and in Special Relativity inertial frames of reference are extremely useful. However, the defining feature of flat spacetime is that there is no gravity anywhere in it.

We can talk about coordinate conditions [2] where those generalize a set of activities that pick out a specific choice of coordinates (and consequently an origin for the coordinates), a choice of units, and some other choices that one can freely make.

The photon's "own reference frame" could be specified as for example keeping it at the origin of a set of flat Cartesian coordinates (x=y=z=0=const) and letting it move against the t coordinate. This is unlikely to be useful, and can be made useless with various choices of units. However, one can say conclusively that the photon's spatial coordinate velocity is zero.

However, coordinate velocity isn't physical: it goes away by changing the system of coordinates. For example, in this system, your coordinate velocity is always exactly c. And so is the moon's. And so is the Andromeda galaxy's. But we can see how unphysical that is simply by fixing coordinates with you always at the origin, or the sun always at the origin, and noticing that the only thing moving with a coordinate velocity of c in those systems of coordinates is the photon.

Indeed, in General Relativity comparing velocities is extremely tricky for objects not occupying the same point in spacetime because it is very easy to be misled by what you're being told by the coordinates and the choices "hidden" within them. The usual advice is to avoid such comparisons (cf. Baez [3]).

Nowadays pretty much every relativist will tell you that Special Relativity emerges from General Relativity (the more fundamental theory) as a special case in the limit where gravity is weak, even though historically Special Relativity came first and informed the development of General Relativity. (They'll also probably advise you to calculate using Special Relativity forms of physical formulae where you can do so!)

However, where gravity is non-negligible or where one is tempted to use a broader set of coordinates than e.g. spherical or Cartesian coordinates on a local patch of sufficiently flat spacetime, Special Relativity is simply inappropriate. Intuitions from Special Relativity about how an object moving at exactly light speed (or even extreeeeeeemely close to it over sufficiently long intervals, like with an extragalactic ultra high energy cosmic ray[4]) are likely to be misleading; instead, one should use the toolset of General Relativity.

Unfortunately that toolset requires complicated mathematics. [5]

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[1] We can define locally inertial frames of reference (LIFs), and the Lorentzian structure of spacetime (four dimensions, three of one sign and one of the opposite sign) guarantees that we can do this in many cases, especially in infinitesimally small regions around a point, or in a small region along a geodesic. (I can explain this further if you are very interested, but its fairly technical and grinding it down to something suitable for HN may take some iteration. I don't even have a link to a decent explanation for what e.g. Fermi coordinates are, why they're useful, and how to use them :( so maybe I'd be breaking new ground ;) ). Some LIFs can be more extensive when gravity is sufficiently weak in that region: an Earth-based "laboratory frame" in Special Relativity is really just a LIF without admitting it; particle colliders typically don't really have to consider the influence of the gravity of bodies like the Earth, the Moon, the Sun, and so on, even if one has a view of the ocean (and its tides) out of one of the lab's windows.

[2] https://www.wikiwand.com/en/Coordinate_conditions goes pretty deep on this

[3] http://math.ucr.edu/home/baez/einstein/node2.html third paragraph

[4] https://www.wikiwand.com/en/Ultra-high-energy_cosmic_ray

[5] where one is dealing with generalized matter (light and massive particles, etc) in general curved spacetime, one might want to take the initial values approach in http://fanfreluche.math.univ-tours.fr/notes/geroch/geroch_no... (or if you like) ch. 27. onwards of https://books.google.co.uk/books?id=NOJ9AgAAQBAJ&printsec=fr... but also in more recent work -- Special Relativity is essentially a framework which has done all this heavy lifting for the user.