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Because doubt is an old word (the OED attests it from 1225) there are a number of nuances of meaning. In summary, with my numbers not OED's:

1. A subjective state of uncertainty about the truth or reality of something;
2. (plural) A feeling of uncertainty as to something;
3. The condition of being objectively uncertain;
4. A state of affairs which gives rise to uncertainty;
5. A matter or point involved in uncertainty; a difficulty (Obsolete);
6. Apprehension, dread, fear (Obsolete);
7. A thing to be dreaded; danger, risk (Obsolete).

As a singular noun, it is decidedly abstract and refers to a state of being. Thus one can be in doubt about something. As a mass noun referring to a state of being, doubt would not normally be pluralised.

When it refers to a particular feeling of uncertainty it is pluralised. Thus if one has that feeling, one has doubts. One does not "have a doubt" because the singular doubt refers to a state of being; senses 5, 6 and 7 which refer to particular things are obsolete in Standard English.

The uses which are obsolete in Standard English may have survived or been resurrected in other dialects. The ELU references say that it does occur in Indian English.