My assumption is that a microchip is like that thing above, not a green board with soldered connections but that it represents what one might put together on a bread board except in a much more condensed form. Is that correct?
Yes, and no.
First, Microchip is a brand name producer of integrated circuits. (Think Xerox, Dry Ice, Thermos etc.). The name has become synonymous, almost genericized, to mean integrated circuits.
Second, an integrated circuit, as pictured, is the result of decades of engineering evolution. Single ICs have replaced what used to be a bigger circuit of various passive and active discrete parts (transistors, resistors, capacitors, etc.). Miniaturization and greater efficiency through the decades has allowed them to become very very small.
But an IC, while being a replacement for a discrete circuit, is also part of larger circuits. And then, newer ICs can integrate those functions as well.
The green board with soldered connections is referred to as a printed circuit board (PCB), often populated with various ICs.