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,](https://golem.ph.utexas.edu/category/2006/09/ringoids.html) and [algebroid](https://en.wikipedia.org/wiki/R-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.