@jahangir ahmed I'm not sure if you're still active here, but I thought I'd provide a brief rundown of what kinds of tests might shed some light on your question.
The first thing to do is to determine whether it's mostly soap (fatty acid salts) or mostly some kind of synthetic detergent (abbreviated here as "syndet"). For this I would take a small amount of the soap and dissolve it in distilled water at a concentration of 1-2%, then check its pH using pH paper or, better, a pH meter. Genuine soap will have a pH in the range of 9-11. If it does, then add some acid (any acid will do, including vinegar) until the pH falls below 6. A genuine soap will convert to its fatty acid form and become insoluble - the solution will get cloudy and particles of fatty acid will be visible. Most syndets however remain stable in mildly acidic solutions and you will see no change.
If it turns out to be real soap, then you will have to find a lab that can do more detailed tests. The main thing you will want to test is the fatty acid composition. This can be done by gas chromatography (GC). The way this works is that the soap is converted into fatty acid methyl esters (FAME, also known as biodiesel) by heating it in acidified methanol. The FAME is extracted with hexane or another hydrocarbon and this is injected into a GC system. The sample is vaporized and goes through a column containing a material that retains some compounds more than others, separating them based on how strongly they interact with this material. A detector at the end of the column measures the amounts of compounds as they come out, and this is used to determine how much of each fatty acid is present, each in the form of its methyl ester.
From the fatty acid composition, you can't determine the exact oils that were used, but you can gain a general idea. The amount of lauric acid can tell you how much coconut or palm kernel oil was used. A large excess of palmitic acid compared to other fatty acids is a sign that palm oil made up a significant amount of the total oils. If linoleic acid is present above about 7%, you can be fairly sure that some liquid oil was present. Detection of longer acids like behenic or lignoceric acids would be a sign that peanut (aka groundnut) oil is a likely component. Keep in mind as well that most commercial soap makers will use only fats that are relatively cheap in your country, so you can narrow down the list of likely components this way as well.
With that information, you can reverse-engineer a formula that will closely match the fatty acid composition. It won't necessarily tell you the exact proportion of different oils that went into the soap, but your soap will behave very similarly to the commercial soap if the fatty acid composition matches it.
