A short answer to the question might be that, after the Nobel award to Einstein in 1922, many people were given the impression that it had been awarded for relativity, and that consequently he received few or no further nominations for any second Nobel prize. I recall seeing a biographical statement to this effect (but can't now locate the reference).
What does emerge from citable records, however, and makes any shortage of further nominations understandable, is that much in the record could well give rise to a mistaken idea that Einstein's prize actually had been awarded to him for the theory of relativity. The following examples would contribute to such a mistaken impression.
First, there was confusion in the media about the subject of the award. When the New York Times announced the award to Einstein, it ambiguously mentioned relativity, but not a word about the actual subject of the award, Einstein's study and results about the photoelectric effect:
"NOBEL PRIZE FOR EINSTEIN. The Nobel committee has awarded the physics
prize for 1921 to Professor Dr. Albert Einstein of Germany, identified
with the theory of relativity, and that for 1922 to Professor Neils
{sic} Bohr of Copenhagen."
-- from New York Times, Nov. 10, 1922, page 4.
Second, confusion also arises from the circumstances of the award and the lecture that followed it. The usual pattern is that Nobel laureates give a lecture about the subject of their prize. There is indeed a lecture by Einstein in the Nobel archives, but it is not about the subject of his prize, it is about relativity: The Nobel Prize in Physics 1921 -- Einstein Nobel lecture (July 11, 1923). The background explains this to some extent. When Einstein received news of his award, he was already on his way to Japan, and perhaps could not attend the award ceremonies in person, at all events he did not attend (Abraham Pais (1982, Oxford UP), 'Subtle is the Lord' : the science and the life of Albert Einstein' at p.503). At the award ceremony on December 10, 1922, Rudolf Nadolny, German ambassador to Sweden, accepted the Nobel prize in Einstein's name and made diplomatic acknowledgements on his behalf -- and naturally there was no scientific content in what was said.
After the Nobel ceremonies, Einstein was invited to Sweden to lecture on a subject 'of his choice'. Arguably this muddied the waters and made confusion worse about the subject of the prize.
"In March 1923 Svante Arrhenius, one of the Committee members, wrote
to Einstein suggesting that the latter not wait until December for his
visit to Sweden but that he come in July. He could then attend a
meeting of the Scandinavian Society of Science in Goteborg on the
occasion of the 300th anniversary of the founding of that city.
Arrhenius left to Einstein the choice of topic for a general lecture,
'but it is certain that one would be most grateful for a lecture about
your relativity theory'. Einstein replied that he was agreeable to
this suggestion, though he would have preferred to speak on unified
field theory."
-- Pais (1982, cited above), p.505.
Einstein, when he finally arrived in Sweden several months after the Nobel award ceremonies, did indeed lecture about relativity. His lecture was not formally linked to the Nobel award, it was given at a scientific meeting in Gothenburg. But it is preserved in the archives among the Nobel lectures, and it is a little difficult to find indications that make clear the relation between the lecture and the prize. There is (now) a fine-print footnote added to the English translation of Einstein's lecture (but not with the original German version). The material is headed 'Nobel lecture' and the footnote to the English version runs: "The Lecture was not delivered on the occasion of the Nobel Prize award, and did not, therefore, concern the discovery of the photoelectric effect."
This combination of events and records was inherently likely to reinforce any confused idea that actually Einstein had already received a Nobel prize for relativity. Mistakes and myths have arisen from lesser causes than that.