Is there a standard classification scheme for different areas of study and research for chemistry like mathematics has Mathematics Classification Scheme (MSC)?
Neither one of the following two might be rigorous enough from the perspective of a mathematician because of some overlap of the fields within chemistry, or by initial inspiration / interaction with fields outside chemistry. Thus I think the answer is no.
However, curricula at universities typically divide chemistry into general, inorganic, organic, physical, analytical chemistry, biochemistry, etc. This is reflected in study guides like chemistry.se's compilation of resources for self-study, or the organization e.g., of bookshelfs Chemistry LibreTexts. As an example of topical overlap however, there are spectroscopic or electrochemical methods which may attributed to physical chemistry as well as of analytical chemistry. A topic like bioinorganic chemistry may be recognized as an offspring in common of inorganic chemistry and biochemistry. There equally are subjects where chemistry may be very relevant, but "chemistry" is not part of the name, too, like toxicology, geology, materials science, to mention a few.
Complementary to this, for long, there have been attempts to classify knowledge into divisions. This is reflected by the labels (not only scientific) libraries put on their media (books, journals, etc.). One of them is the system established by the Library of Congress with starting letters
QD as root for chemistry within science (cf. entry wikipedia, or entry LOC). An example for this may be the record for Chemistry of the Elements by Greenwood and Earnshaw here and other topically related entries of shelf QD466.
An other attempt are IPC as the international and CPC as the cooperative patent classification (USPTO and EPO) assigning patents related to chemistry (and metallurgy) to class
C (entry EPO/espacenet). This classification however may be less helpful than a library catalogue à la LOC since patents may be filed in more than one (sub-class) at once. Equally, you may find multiple patents related e.g., to keyword thiophene in classes other than