you can claim, but it is difficult due to chances of high rate of mutation rapidly in case of microorganism. Therefore, your primers reproducibility will be decreased rapidly, even you do the experiment by collecting the same microorganism outside of your lab.
It is very complicated to claim that your primer is specific for your microorganism. The problem is that how can you be sure that this primer can not work or identify all other microorganisms distributed in the universe. Many claimed primers designed for specific pathogenic bacteria are found to identify other non pathogenic microflora naturally found in soil.
The answer to your question, in such a case, you can claim that your designed primer is specific for targetting specific microorganisms in pure culture.