I’m trying to find the term (if any exists) for this:
What is it called when the size of individual or groups of words are increased proportionally in size so that they are all the same width?
Just changing the size to make the width the same but the height different is simply called “typesetting”.
Making sure that multiple lines (read: a paragraph) fit left-aligned, right-aligned, center-aligned, or block-aligned is called “justification”.
But since some of your lines only contain a single word, the terms you are probably looking for here are “kerning” and “tracking”. “Kerning” adjusts the space between individual letter forms — in contrast to “tracking” (letter-spacing) which adjusts spacing uniformly over a range of characters.
Both “kerning” and “tracking” can be used to tweak the widths a bit better than just resizing. When you use a text-editor (or software like Photoshop) and justify your paragraphs, kerning and/or tracking do the magic.
Other options and terms for other (alike) options to influence the width of your text would be “word spacing” and “sentence spacing”, but in your question’s example, those are obviously not used.