It‘s an Arduino Uno, but the specific model probably isn‘t what you were asking for
As for where to start, you should probably start with Processing! Just create the Code and see if it works with mouse/keyboard Input. Once it does you can think about exporting the input to a Controller or somesuch.
Anyway, i‘ve got little experience with using Controllers other than arduino with Processing, so don‘t put too much weight on my words, but i‘d say using Arduino is probably a lot simpler than using a controller of some other kind.
Apart from drivers/libraries and other things you might need, there‘s always the risk that some might not have been updated for years and don‘t work anymore.
You probably won’t have that Risk with arduino, since everything you‘d need for this project is the arduino serial library. It‘s pretty easy to use and relatively up-to-Date.
I don’t know much about font making but I’m pretty sure that “Parametric Font” is the term coined to describe what you are trying to achieve. Googling that expression might help finding further information regarding the creation process. Also, note that there are websites like Modulator that make possible to create / morph fonts online and download them without having to use code.
Now regarding the website you linked, I could glean the following information:
Morphing / animations seem to either be made with Illustrator (IG post) or in Processing (short video)
Fonts are most probably made with Glyphs, a popular font editor. The short video above shows code within a Processing sketch (you can clearly see the setup(), draw(), background(), keyReleased() and keyPressed() functions) that is referring (not calling) to functions or parameters from the GlyphsApp API ( OFFCURVE, nodeList, showInterface).
The name of the sketch "Glyphs_Data_Import_Test" suggests they retrieve the fonts data with the API first and then import it in a Processing sketch for manipulation.
If you are looking for a free (and possibly less elaborate) alternative you can check the Geomerative library that allows, among other things, to access the path of a font ( its “skeleton”, .ttf format only) and manipulate it as you wish. In addition to the bundled examples, you can find interesting sketches browsing the old forum or looking at chapter P.3 of Generative Design.