The book "Математика в СССР за 40 лет 1917-1957" (Mathematics in USSR in 40 years 1917-1957) mentions what seems to be both relevant Rabinivitsches. This is not definite, though, since the bibliography only has papers published in USSR.
One is Ю. Г. Рабинович (Yu. G. Rabinovitsch), the book only mentions two of his papers, in 1918 and 1921. This is probably Rainich.
Another is Юлий Лазаревич Рабинович. That can be transliterated either as Juli L. Rabinivitsch, or as Yuli L. Rabinivich, so it is consistent with the initials in Math. Ann. paper. The book only mentions his papers from 1948 to 1957, and those are mostly in Functional analysis, PDE and Special functions. He was born in 1894 in Bobruisk and graduated from Moscow State University in 1924, where he worked since 1932 and was awarded a Ph.D. and a title of Docent in in 1935. So, the bio is also consistent with him being research-active in Moscow in 1929.
Update: his birth and death dates are 1.09.1984 - 18.03.1968 (the former is from the above-mentioned source, and the latter from the preface to the 2nd edition of Courant-Hilbert that he translated with Libin.) I found mentions of J. L. Rabinovich in several memoirs (including one mentioned by Carlo Beenakker below) as a docent in the Physics department of the Moscow State University. He was remembered as a brilliant lecturer.