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.
The line length is referred to as the "measure." The above sample of different words on different lines are set "flush." They may also be referred to as being set optically aligned flush. Flush-left describes the alignment of the left edge of the type block. Flush-right for the right edge alignment. Irregular line lengths can be centred over one another, too. There are a number of irregular variations.
The term to express simultaneous alignment of the left (beginning) and the right (ending) is "justified." It is usually accomplished by varying the word space on each line and by using hyphenation to split multi-syllable words into manageable chunks. Irregular word spaces make reading more difficult than regular gaps between the words. Justified is an appearance, a condition. There are different actions a typographer takes to create a justified text block.
The technique used in the above example is by adjusting font size to fit the line measure. Other means can be used to achieve alignment.
Letter-spacing can be increased to stretch the length of the line to the desired measure. A constant amount of additional letter spacing is also called tracking by some software.