TL;DR: No, a Mage Hand does not move through creatures or objects.
One of the key principles of 5e rules is that there are no hidden rules.
As per @illustro's answer both the Specter and Ghost have a special movement mode:
Incorporeal Movement. The specter can move through other creatures and objects as if they were difficult terrain. It takes 5 (1d10) force damage if it ends its turn inside an object.
Incorporeal Movement. The ghost can move through other creatures and objects as if they were difficult terrain. It takes 5 (1d10) force damage if it ends its turn inside an object.
The Mage Hand spell, however, has no such exemption:
A spectral, floating hand appears at a point you choose within range. The hand lasts for the duration or until you dismiss it as an action. The hand vanishes if it is ever more than 30 feet away from you or if you cast this spell again.
You can use your action to control the hand. You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial. You can move the hand up to 30 feet each time you use it.
The hand can't attack, activate magic items, or carry more than 10 pounds.
In the absence of exemption, the more general rules apply, and the general movement rules do not allow one to move through other creatures and objects.
Ergo, a Mage Hand cannot move through other creatures and objects.
As for the use of spectral, it follows that as Educorreia mentioned it only concerns the appearance of the hand.
Specifically, I would expect the Mage Hand to be see through, as per the illustration from fandom.com:
(Original Source = Xanather's Guide to Everything, p. 127)