Who started the "-oid" suffix fashion in math? There are lots of structures which have name suffixed by "oid".  Off the top of my head, matroid, greedoid, perfectoid, causaloid...
Who started this? AFAIK, "matroid", by Whitney, was a start, and led the way to several combinatorial oids.  However, the Cardioid has had its name for some centuries now, so the use of the suffix is old.
Still, it seems a bit different to name a family of specific objects, and to name some sort of abstract structure.
 A: The suffix "-oid" means the same as "quasi", so "resembling", "like". A groupoid is a quasi-group, like a group. There are hundreds of words in that category, covering many scientific disciplines.
In the "early use of mathematical words" database I find:
250 BC: conchoid
200 BC: cissoid
400: trapezoid
1650: trochoid
1672: ellipsoid
1685: cochleoid
1830: epicycloid
1836: paraboloid
1837: strophoid
1844: centroid
1872: geoid, gyroid
1878: nephroid
1879: deltoid
1881: prismatoid
1891: cuboid
1935: matroid
The Woid on-Oid by William Safire comments on the proliferation of -oids:

We all know that the use of -oid to create a noun has been growing by
leapoids and bounds. Among the earliest were android, or "automaton
in human form," created in 1727, and asteroid, "small body like a
star," in 1802. Scientists and mathematicians were especially
attracted to the ending, juggling their cylindroids, globoids and
spheroids.

A: 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 RMod


C*-algebra
C*-category
*-category enriched in Ban




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.
Also, it should worth remind that not every enrichment of a categories consider as an "odification". For example (Lawvere) generalized metric spaces is a  category enriched in the monoidal poset category $([0,\infty],\ge),$ where the monoidal product is taken to be addition.
