Hi,
The stuff that turns liquid, melts, because you heat is called is "solder." When it cools it gets hard again.
Solder either has resin in the center (resin core), or not (solid). Generally I think resin core is better for hand wiring a keyboard.
Solder joins other metal parts together.
"Wire" is the stuff that never melts... it stays strong and bendable. Wire carries the electricity from place to place. Wire does not have resin.
So, it sounds like you are asking about "wire." Thick wire is used for lots of electricity. Keyboards do not need lots of electricity.
For a teensy 19ga to 22ga will probably do fine. (0.91mm to 0.64mm) 2mm seems thick to me for Teensy and keyboards.
Hope that helps.