Do Vedas say that God is unique? How many Gods are there acc. to Vedas? What are his characteristics by which one says that God is unique?
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Qualities of "God" are described in dRg dRzya viVeka by Sankaracharya
asti bhAti priyaM rUpaM nAma chetyaMshapaJNchakam.h |AdyatrayaM brahmarUpaM jagadrUpaM tato dvayam.h || 20 ||
Everything has five factors: 1) Existence, 2) Consciousness3) Bliss, 4) Name, and 5) Form. Of these, the first three are of the nature of Brahman and the last two (Name and Form) belong to the world.
khaMvAyvagnijalorvIshhu devatiryaN^.h narAdishhu |abhinnAssachchidAnandAH bhidyete rUpanAmanI || 21 ||
In the elements, space, air, fire, water, and earth, in Gods,animals, and humans, (and other things in the world) what isnon-different (constant, unchanging) are Existence, Consciousness,and Bliss. What are different (among all these things) are theName and Form aspects.
Note 1: sat-chit-ananda is synonymous with asti-bhati-priyam, the latter being the corresponding "cognizable attributes" of the former. This is nirguna brahman.
Further, bhagavan is explained as follows:
it is described by Parāśara Muni that
aiśvaryasya samagrasya vīryasya yaśasaḥ śriyaḥ jñāna-vairāgyayoś caiva ṣaṇṇāṁ bhagam itīṅganā (Viṣṇu Purāṇa 6.5.47)
Bhagavān means who possesses these six opulences in full: all riches, all strength, all influence, all wisdom, all beauty, all renunciation.
These are the qualities of saguna brahman.
Note 2: Here aiswarya has been given a rather limited translation. In this context, the quality of being "Ishwara" i.e., lord and master (of all things - material and spiritual) is aishwaryam.
As for the question on how many gods do we have according to the vedas, someone who has read the vedas must answer. The upanishads, which are the concluding portions of the vedas, severally propound one nirguna brahman .