The power is not about being the only one who can use it, it is just that no one can copy, mimick or absorb anything of the users of Singularity. For example imaging there are three people, Person A, Person B and Person C.
Now let's say both Person A and Person B have Fire Manipulation, Person B also has Singularity and Person C has Power Replication and Shapeshiting.
Now, if Person C tries to user their powers on Person A and Person B, it would only be successful on Person A.
Unfortunately for your explanation, Singularity is not the power described by the OP. Singular Power, as they call it, would be the same concept as Singularity, but applied to the user's power(s) rather than to the user themselves, in that it would prevent anyone else from having the same powers as the user (think the Imprison move from Pokémon, but with a permanent effect).
Going off of and applying the correction to your explanation in that Person B has Singular Power instead of Singularity, if Person B had both Singular Power and Fire Manipulation, then it would prevent Person A from even having Fire Manipulation in the first place.
And if Person C tried to copy Person B, they could only copy their appearance since, once again, Singular Power would keep Person C from gaining access to Fire Manipulation.