Do Black Pepper and Black Nightshade share specific characteristics?
Short answer
Nigrum is Latin for 'black'. As you write in your post, it's black pepper and black nightshade. Not until the level of Class and Phylum do they share the same taxonomic classification. Hence, the species suffix nigrum doesn't tell you anything about their kinships, other than their physical appearance (they're both black). 

Background
In general, the species part of the linnaeus nomenclature has nothing to do with evolutionary relationships. Instead, it is the higher classifications that denote kinships, for example, as you already suggest, the genus name, that may have various related species in it (Fig. 1). The species you mention are of a different genus.


The Species: Piper nigrum belongs to the Genus Piper, Family: Piperaceae, Order: Piperales, Class: Magnoliopsida, Phylum (Division): Magnoliophyta, Kingdom: Plantae, and Domain: Eukarya
The Species Solanum niger belongs to the Genus Solanum, Family: Solanaceae, Order: Solanales, Class: Magnoliopsida (Dicotyledons), Phylum (Division): Magnoliophyta, Kingdom: Plantae, and Domain: Eukarya.


They share only the same Class (and hence Phylum, Kingdom, Domain). For comparison, in the Kingdom of Animalia, we have the Phylum Chordata (vertebrates), that splits up into the Classes Mammalia (Mammals), Actinopterygii (Bony Fish), Chondrichthyes (Cartilaginous Fish), Aves (Birds), Amphibia (Amphibians) and Reptilia (Reptiles). So your two plant species share as much in common at the taxonomical level, as Mammals and bony fish.

Piper nigrum (or black pepper), is a climbing vine in the Piperaceae (pepper family) native to the Malabar region of southwestern India. Black pepper is grown in various tropical regions, including India, Indonesia, and Brazil, as it is one of the most important spices in the world.

Solanum nigrum (or European black nightshade) is a species in the Solanum genus, native to Eurasia and introduced in the Americas, Australasia, and South Africa. It is also known as black nightshade, duscle, garden nightshade, Indian nightshade, garden huckleberry, hound's berry, petty morel, wonder berry, small-fruited black nightshade, or popolo. 


Fig. 1. Linnaeus nomenclature