As with many terms in BDSM the phrase collared slave has varying meanings and implications but is mostly used to refer to either the literal case of a slave who wears a collar or more figuratively for a slave who has an owner, rather than someone who is saying they are a slave as a way of stating their orientation, even though they do not have an owner.
Slave with a collar
In historic times of legal slavery, no one who was not owned would tolerate having a collar on them. In those days, not all slaves wore collars. Therefore, we come from a history where slavehood was more likely than being collared and it is still the case that not all consensual slaves wear a collar, or even have done. Many slaves are naturally proud of being considered good enough to be owned and cared for and are therefore proud to display symbols of their enslavement, and hence will make a point of being a collared slave.
Collars are not confined to slavery. Quite apart from vicars, many other people, girls especially, wear collars as fashion accessories. Sometimes it is in this context that a person might want to make it clear that their collar is not something trivial and therefore will state that they are a 'collared slave'.
A simlar phrase to the figurative 'collared slave' is owned slave, which is tautology since a slave is a person who is owned.
These phrases have meaning in the context of matchmaking sites that are targetted for BDSM where it is common for the site to ask people to classify what they are looking for in terms such as 'Master', 'Dominant', 'Submissive' and 'Slave'. Someone who wants to be a slave therefore selects 'slave' as a classification, even though they are not owned. For people trying to make it clear (for example, on forums associated with such a dating site) that they are not a want-to-be-slave but actually have an owner, they might choose to say they are a 'collared slave' or an 'owned slave'.