The explanation for the similarities that you observe is due to the fact that Ahirani derives from something like Sanskrit (a dialect of, or sister language). The Prakrits are probably well-enough attested that you would be better of comparing Ahirani and one of the intermediate languages like the Apabhraṃśas, in the same way that one could compare Latin and French (knowing that French descends from Latin -- so this isn't standard comparative linguistics where one compares the known Algonkian languages and hypothesizes what the original language looked like). However, before you can compare two Ahirani and anything else, you would need to have some materials on the language. Ahirani-shabdkosh seems to be large enough that you might actually be able to put words of the language next to words of some older language.
It might be useful to first read some of The Indo-Aryan Languages edited by Cardona and Jain, which would put the development of the languages into perspective (it would tell you what is already known).
One problem in understanding the development of any modern especially Indo-Aryan language is that languages can freely borrow from each other. Prakrits borrowed from Sanskrit as well as retaining original words which went through centuries of language change; Apabhraṃśas did likewise, and modern languages can borrow from earlier (literary) languages as well as modern languages such as Hindi and Gujurati. Romance languages often have words that descended in the normal way from Latin (for example Spanish leche "milk"), as well as words borrowed later from Latin (such as lactar "to lactate"). In order to know whether you are dealing with borrowing or direct descent, you have to know the regular historical rules. If a language has a word like ishwar for "god", that is almost certainly a later borrowing from Sanskrit, because shw was simplified millenia ago.