Yes, there is a standard Mercator projection. However, as with (almost) any other projection designed for the earth's surface, it can use different ellipsoidal models and has different aspects:
The ellipsoidal model describes an idealized shape (and size) of the earth. Most projections have formulas that apply to any ellipsoid of rotation, whose shape is determined by its "flattening" (or an equivalent parameter such as the eccentricity).
The aspect describes the orientation of the ellipsoid. The standard Mercator projection orients north to the top and is centered at the equator and prime meridian. You can vary these three parameters by applying the projection formulas to the rotated ellipsoid. This is not straightforward when the ellipsoid is not a sphere, because if you rotate the axis away from north-south the ellipsoid's shape with respect to the plane on which it is projected actually changes slightly. However, some rotations are relatively easy to handle: any rotation about the axis just shifts the meridians east and west. A rotation that places the axis along the (former) equator--that is, a 90 degree rotation--is usually easy to handle. Such an aspect is called "transverse." All other aspects are known as "oblique."
Therefore, in choosing any projection, you usually get to choose not only the projection itself, but also the earth's shape and the aspect of the projection. The Mercator is commonly used in its standard (equatorial) and transverse aspects with either a sphere or a slightly flattened spheroid (such as the WGS 84 spheroid). It is rarely seen in an oblique aspect. No single one of these choices is universal: as always, the decision depends on your accuracy needs and on how you plan to use the projection.