Yes, Spirit Guardians' area depends on the size of the caster
Spirit Guardians has a range of self and the area of effect is described:
They flit around you to a distance of 15 feet for the duration.
A distance of 15 feet is the same range that you could hit if you have 15 feet reach or a ranged weapon with 15 foot range. Unless you are using the variant facing rules you are assumed to have that reach in all directions from all squares that you occupy.
Therefore Spirit Guardians' area of effect is the area of a 15-foot reach from the caster. Assuming the caster is Small or Medium, this is slightly larger than a 15-foot radius, as it adds the square the caster is standing in. For a smaller creature that does not occupy a full 5-foot square, this is the same area as a 15-foot radius. For a Large or larger creature, the area is much larger.