I assume you mean the height adjustment tool that allows you to alter the height above the ground of a scenery object, such as a person - and not the terrain height adjustment tool? Then the answer is no (it is also no for the terrain height tool as well).
The default height setting for most objects is ground level and the height adjustment tool starts at that level. Some scenery objects have their default height set at ground level + an offset, usually the height of a platform. For example a platform seat may have an offset of +0.65m which means it will fit perfectly on a platform of that is 0.65m above ground level so you can place it on the platform without having to increase its height. This offset is added by the objects designer.
The problem you would have would be that buildings and other objects could, and probably would, have different heights and so would need different default height settings anyway.
