Following the comment by Alfred Centauri, let me suppose that you discussed the Dirac theory of electron, and not the Schrödinger one. I will come back later on the possibility to describe matter-light interaction using the Schrödinger equation.
The Dirac theory describes the (special) relativistic behaviour of a particle. When complemented by the principle of gauge invariance (in particular the substitution of the normal derivative by the covariant one), it gives the basic playground for the simultaneous descriptions of the electromagnetic field (Faraday law and absence of magnetic monopole), the charge associated to the relativistic particle (the equation which replaces the Newton equation with the Lorentz force if you wish, but this has to be though with care) and their coupling (equations similar to the Maxwell-Ampère and Gauß, but there the current and charge densities have full quantum meaning, no more fluid interpretation as for the classical electromagnetism).
Obviously everything get more complicated when you try to quantise the electromagnetic field. The previous discussion didn't discussed the appearance of the photon.
I would say that the Wikipedia page related to the Dirac equation is not so helpful for understanding this point, but you could try to open the book by A. Messiah Quantum mechanics (volume II if not in an edition with the two volumes in one book), which contains all the pedagogical details you need, including the quantisation of the electromagnetic field in term of photon.
Schrödinger vs. Dirac description of matter
One can also describe the interaction between matter and light using the Schrödinger equation for the atom. This is the main study of the book
C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg Photons and Atoms: Introduction to Quantum Electrodynamics, Wiley (1992)
that I suggest you to read. In short, when the magnetic-like interaction is weak, the description using the Schrödinger equation is sufficient. You can understand this with the pictorial idea: when the (continuous laser) light field does not interact too much with the (gas) atoms, the effect can be describe by the first order term in interaction, which is already given by the Schrödinger prescription.
Now, regarding your historical perspective, it seems highly not probable that Dirac would have described the coupling between electrons and photons if he were unaware of the Maxwell's equations. This is once again because the gauge invariance is crucial in deriving the coupling. You may find more details about the history of gauge theory in the excellent collection of historical articles by
L. O'Raifertaigh The dawning of gauge theory, Princeton series in Physics (1997).
The same reasoning apply to the Schrödinger equation, because all these physicists were deeply influenced by the notion of field, that Maxwell really invented half a century before.
In short, the gauge invariance is the main ingredient of matter-field interaction, not the equation you're using to include it.
To be also noted:
The particle behaviour of light was not the generally accepted behaviour of light (as you said) at the time of Maxwell's equations. Indeed, the Young two-slits experiment was already known by the end of the 18-th century.
You do not really need to quantise the photon field to understand the photoelectric effect. This is discussed in a paper
Lamb, W. E., & Scully, M. O. The photoelectric effect without photon, in Polarisation, matière et rayonnement (pp. 363–369). Presses Universitaires de France (1969).
where they calculate the photoelectric effect quantising only the electron / detector behaviour.