Will the temperature of a faucet actually be lower than the countertop or will it only feel cooler due to specific heat?
The short answer: 

A metal object as the faucet drains heat from fingers due its high thermal conductivity faster than the countertop.

The fingers get therefore colder than if touching the countertop.

The faucet perceived temperature is therefore colder, even if the faucet and countertop temperature are the same.

If it is really colder due cold water, the perceived coldness is even stronger.

The longer answer: 

Metals have high thermal conductivity.
That causes higher thermal flow at the same temperature difference than for most non metal materials of the same temperature. ( Rare exception are diamonds, as their thermal conductivity is superior to all metals.).

That leads to perception like if metals were colder ( if being colder then body temperature ) or warmer ( if being warmer ) than they really are.

OTOH, metal (mass-)specific heat capacity  is mostly lower than for non metals, due heavy metal atoms, that in fact somewhat decreases the effect of high thermal conductivity.

For your specific case, it depends also on the real metal temperature, e.g if it is partially sharing temperature with cold or warm tap water.

Similarly, water feels like colder than air of the same temperature, as thermal flow from human body is higher. But at this time, the higher volume specific thermal capacity plays major role.



2 examples how our senses can be deceiving:

If you had not too much hot herbal tea in an aluminium cup, you could still be sipping the tea itself without burning your tongue, but the cup would feel like burning your lips. If you did the same with the stainless steel cup, it would not be feel like burning as much, but still hot. If we used thermometer sensors, we would realize the metal surface not in direct contact with tea is very slightly colder than tea. But in both cases you would have impression the metal is warmer than the tea itself.

This time our senses monitor real thing ( temperature ), but rather their own temperature, instead of the temperature  of the sensed object.



Let have 3 basins with cold water, lukewarm water and warm water. You measure their temperature with thermometers, so it is a real thing. Now you put your left hand to cold water and your right hand to warm water for a minute. Then you put both hands to lukewarm water. The left hand will give you perception it is warm water, while the right hand will give you perception it is cold water.

This time our senses process not the temperature itself, but rather the temperature spacial and temporal gradient.