I know that Sodium has high polarising power than potassium so it can convert the respective functional groups more readily than Potassium and both Lithium and Sodium salts are soluble in water. This indicates that both Sodium and Lithium can be used but this is not the case as in each case the organic compound is first fused with Sodium to make respective ionic compounds and then identification tests are done. Why is this so? Why any other metal is not used?
You asked this historical question before (check Sci math history which is pretty good). For historical searches, Google scholar works the best. https://scholar.google.com/scholar?hl=en&as_sdt=0%2C44&q=lassaigne+test&btnG=&oq=lassaign
The very first article is very useful on the history of sodium fusion test. "A lost centenary: Lassaigne's test for nitrogen. The identification of nitrogen, sulfur, and halogens in organic compounds" J. Chem. Educ., 1945, 22 (5), p 212
The first few lines of the article clearly mention that Lassaigne used potassium rather than sodium. He must have the same question his mind which you are asking now. Surprisingly, it was another person named Jacobsen who suggested using sodium. The original test indeed used potassium.
Why sodium is used today in some teaching labs? Lower price and less reactivity than potassium. Essentially nobody uses this historical test in analytical laboratories anymore. Today, an very small quantity of organic compound is burned and the gaseous products are analyzed by gas chromatography.
Historical availability and procedural compliance. Why to use potassium or lithium if the procedure says sodium ? Plus, lithium is significantly more expensive and reacts with gaseous nitrogen. Potassium may be too reactive for manipulation and reaction with some samples.