In asexual organisms such as bacteria, archea and some fungi, as well as in some plants where asexual reproduction is the only reproductive strategy, how can we be unambiguous in defining if an organism belong to a different species from another or not?
There is no taxonomic category called "asexual organism". It is rather a phenotypic trait of scattered occurrence.
However the absence of sexual-reproduction in a group of organisms sometimes causes problems in scientific classification. This caused development of the artificial-group fungi-imperfecti (or deuteromycota) at one time, for fungal strains which lack sexual reproduction.
Bacteria and archaea (prokaryotes) are not known to show sexual reproduction. However they show some gene transfer processes like conjugation and transformation; but this is not the same as sexual reproduction because cell fusion does not take place.
Defining a species or describing how one species differs from another goes beyond an animal's ability to mate with another. I get the idea that's where you were heading with this train of thought and you're not entirely wrong. An early technique of classification, developed by Carl Linnaeus, relied on the number of stamens a plant had to define a species. Differences in reproductive organs, mating behaviours and gametes are certainly characteristics that can and do help taxonomists to categorize organisms.
Modern taxonomy takes a number of characteristics into account and there a number of different branches of taxonomy that focus on a certain set of characteristics.
Many forms of taxonomy rely on shared descent to classify organisms based on shared common ancestors. This is called phylogenetics. Taxonomy also uses shared characteristics, rank-based classification and genetics to define species.
Asexual species can also be classified with these techniques but there some challenges associated. Sometimes asexual species have rapid life cycles (eg. bacteria) and therefore evolutionary change can occur rapidly. Horizontal gene transfer can occur transmitting DNA from one species to another and makes the process of classification tricky.
That being said, evolution is a constant process and genomes are constanly changing. Classifying any species that are continually changing is challenging.