A counsellor can do the job much better than a chaplain who is clearly a member of a specific belief system
Chaplains are specifically trained
in multiple religious practices, and to help service members in their religious activities, even if these activities are not the chaplain's own.
Some of the services I attended on military bases through the years were, to be honest, quite generic.
Imagine a bajoran vedek having the chaplain job on a ship, could someone from earth relate to him?
If the Vedek were trained as a chaplain
, then yes. To assist a particular Starfleet member with their religious need, the Vedek might have to briefly excuse themselves and consult reference materials. The Vedek would as part of their chaplain training, be instructed in basic religious concepts outside their own beliefs.
In time the Vedek would come to know the members of the crew, and the details of their religious needs. As I stated earlier, not all religion is do it yourself.
Kind of be hard to do your own Bris.
And how would this be different than one of the doctors (except the EMH) having to consult medical references, prior to treating a species that was known to the Federation, but they themselves had never treated before?