So, i was reading about the origin of life, and i've understood that one of the reasons why we believe there was a LUCA, is due to the fact that there are a lot of characteristics common to all organisms currently living on Earth. Then i've thought that maybe, the use of comparative genomics has already proven that certain characteristics, e.g. the existence of a DNA polymerase, have had the same origin. I don't really know how comparative genomics work, but i'm assuming that they compare the DNA sequences that code for a certain protein, and if they have a relatively low ammount of mutations between the different organisms, then they can assume that the DNA sequence came from one inicial organism. In this train of thought (i might be wrong about it though) i was wondering, is the DNA sequence (the gene) that codes for DNA polymerase (at least for the same type of DNA polymerase, i know that various types exist) the same between, for example, a mammal, and a bacteria?

No, even highly conserved genes we you will find sequence differences and they generally scale with the distance between organisms. Specifically for DNA replication there are marked differences in the structure and replication between eukaryotes (such as us) and prokaryotes (such as bacteria) and require different polymerases to work.