Expanding upon my comment: categories without units are called semicategories. You can enrich a semicategory in a semigroupal category, which is what you describe. The Yoneda lemma is subtle with semicategories, but see On regular presheaves and regular semi-categories. However, note that the authors work with semicategories enriched in a monoidal category: this is because, despite the definition of enriched semicategory and semifunctor not needing a unit in $\mathcal V$, a unit is necessary to define an enriched notion of natural transformation between semifunctors. I am not aware of a reference that explicitly develops the theory of semicategories enriched in semigroupal categories.

Expanding upon my comment: categories without units are called semicategories. You define a notion of semicategory enriched in a semigroupal category, which is what you describe. The Yoneda lemma is subtle with semicategories, but see On regular presheaves and regular semi-categories. However, note that the authors work with semicategories enriched in a monoidal category: this is because, despite the definition of enriched semicategory and semifunctor not needing a unit in $\mathcal V$, a unit is necessary to define an enriched notion of natural transformation between semifunctors. I am not aware of a reference that explicitly develops the theory of semicategories enriched in semigroupal categories.