Though this might not be what you are expecting, I will explain you "oidification" or horizontal categorification as I understood (Experts are fell free to add or edit as necessary). This is the process that generalizes a "certain type of category with a single object" to "such type of categories with multiple objects". This is done mostly via "enriching" the initial category $\mathcal{C}$ over another monoidal category $\mathcal{K},$ which roughly says arrows of $\mathcal{C}$ are replaced by objects of $\mathcal{K}.$
Examples includes \begin{array}{|l|l|l| } X & X-\text{oid} & \text{}\\ \hline \text{monoid} & \text{Category} & \text{ categories enriched over } \mathbf{Set} \\ \text{Category} & 2\text{-Category} & \text{ categories enriched over } \mathbf{Cat}\\ \text{Group} & \text{Groupoid} \\ \text{Ring} & \text{Ringoid} & \text{category enriched in the tensor category } \mathbf{Ab}\\ \text{Algebra} & \text{Algebroids} & \text {category enriched in } \mathbf{Vect} \text{ or } R\mathbf{Mod} \\ C^{\ast}\text{-algebra} & C^{\ast}\text{-categories} & \ast\text{-category enriched in } \mathbf{Ban} \end{array} You can find more details ringoid, and algebroid here. But as far as I know Hopf algebroids and Lie algebroids does not fit into this general definition of algebroids, but still multi-object generalizations of their counterparts.