What are the theological characteristics of an Idol or Statue to be considered as Object of Shirk?
If those Characteristics of an Idol is also found in some other object , then does that object also become an object of Shirk?
Shirk is simply the act of attributing partners to God, or giving/attributing to others that which is for God alone. An "object of shirk" is just something that someone commits shirk to; this could quite literally be anything.
Shirk is also hardly limited to idols or statues; early peoples would worship all manner of natural objects, including but not limited to the sun, the stars and trees. Even Christ Jesus himself is an object of shirk to many Christians (although he will rightfully deny having had anything to do with that).
On the other hand, the idols of the people Nuh weren't objects of shirk at all when they were built; people did not start worshipping them until multiple generations had passed.
There really is no defining characteristics that tie these objects together except that they're objects which man, in his ignorance, chose to worship instead of (or in addition to) God.
In addition to the view proposed by the other answer, I found some distinctive, yet not defining characteristics of an object that would cause people to worship it , if not now then in future:
Allowing the disbelievers (like pagans) to worship that object
An example of this characteristic is the Shrines or Dargahs which are found in the Indo Subcontinent which is inhabited by a majority of the Muslim world. These Shrines\Dargahs are promoted to the disbelievers and the disbelievers from all religion flock to the Shrine for their petitions and supplication to the object(grave) housed in the Shrine. Even the deviant Muslim sects encourage other Hindu people to perform worship in their own pagan way like offering food to the object, making circumambulation of the Object etc.
As shown above a famous Hindu women is allowed and escorted to worship the Shrine ( Dargah ) of Khawaja in Ajmer, by offering clothes and offerings to be carried on head in dedication to the object housed in the Shrine.
Making the Object very magnificent and larger then life size
Its a common characteristic seen in various Idols used by the disbelievers to make it magnificent or very large in size like the Buddha statues or the alleged Shrine of Imam Ali A.S. in Najaf Or the most notorious Shirk promoting Shrine in the world named as Khawaja's Dargah in Ajmer, India:
Dedicating the Object to a creation rather then the creator(Allah)
Various objects that have become objects of Shirk are all dedicated in the memory of other then Allah swt. The initial intention in building such objects would be as a memorial to the person representing the object but as time passes by people start attaching special significance to that Object and dedicate it to other then Allah swt.