What is it called when a word is abbreviated from the middle? (Ex: fridge from reFRIGErator)

Stuart and Edwin have identified the term "clipping". One example that's coming to mind is "still" when used as a noun for a piece of equipment used in distillation.

But there's some discussion that this might not be happening with "fridge". Another possibility is that it's from the first part of the name "Frigidaire", which suggests a U.S. brand-based origin (and which has that "d" in it), though the first link mentions at least one instance of " 'friges' " (with apostrophes or possibly single quotes) before the name was adopted by the company.

This Wikipedia article attests:

Word formation/Clipping

Clipping occurs whenever a word is shortened but nevertheless still retains its original meaning and belongs to the same word-class.

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There are four types of possible clipping processes, depending on which part of the word undergoes structural changes:

• back-clipping (temperature — temp, rhino — rhinoceros, gym — gymnasium),
• fore-clipping (helicopter — copter, telephone — phone, plane, aeroplane),
• mixed clipping (influenza — flu, refrigerator — fridge) [Elizabeth — Liz, and (after Crystal) (from the compound) french fried potatoes — fries],
• clipping-compounds (paratrooper — parachute + trooper).

It seems to merge medial clipping in the superset 'mixed clipping'. But this is obviously the correct set for the abbreviations involved here.