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 homsets (set of arrows two objects) of $\mathcal{C}$ are replaced by objects of $\mathcal{K}.$

Examples include 

| *X* | *X*-oid | Enrichment |
|-|-|-|
| monoid | Category | categories enriched over **Set** |
| Category | 2-Category | categories enriched over **Cat** |
| Group | Groupoid | |
| Ring | Ringoid | category enriched in tensor category **Ab** |
| Quantale | Quantaloid | category enriched in suplattices |
| Algebr | Algebroid | category enriched in **Vect** or *R***Mod** |
| *C*<sup>*</sup>-algebra | *C*<sup>*</sup>-category | *-category enriched in **Ban** |

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.