Expanding upon my comment: categories without units are called [semicategories](https://ncatlab.org/nlab/show/semicategory). 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](http://www.numdam.org/item/CTGDC_2002__43_3_163_0/). 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.