I often recommend the book 'Cats' Paws and Catapults' by Steven Vogel, which discusses the differences between human and biological engineering. There are very different constraints, as well as the more obvious difference between intelligently directed design (by humans) and evolutionary design.
It's not totally true that our engineering is simpler than biological systems. After all, we can now manipulate matter down to the atomic level. There have been attempts to move towards self-assembling systems, or even self-repairing ones. Not quite on the same scale or sophistication as biological systems, of course.
Is a modern computer chip more complex than a virus? Is a large connected system of computers more complex than a mycelial network? Are there more parts connected in more ways in a nuclear power station as there are in a simple bacterial cell?
I think it is definitely worth considering the similarities and differences.
I think my sense that biology is more complex than human engineering is that our engineering seems much more homogeneous. Computer chips are certainly very dense but the individual number of different component types are very small. Biology on the other hand seems to have a huge number of fundamental building blocks, e.g. proteins, with different behaviors. I suppose that maybe that's the wrong comparison though and software is where our engineering diversity is present.
It may well just be the case that my lack of familiarity with biology makes it seem more complex to me.
But nobody will ever understand everything about a natural cell. Levels are an ad hoc phenomenon adhered to in varying degrees in certain places to contain variation. But few elements have only one role at one identifiable level, and you can never be sure one doesn't have another. And huge amounts of apparatus might radically change behavior in a subtly different environment.
Software is where the diversity is, for computers. A cells systems are a set of interlocking networks of DNA, RNA, protein, small molecules, and signalling. As you say, these diverse parts then interact in diverse ways.
Another aspect is the large number of 'levels' in biological systems - humans have around seven; atoms, molecules, organelles, cells, tissues, organs, whole system. The interactions within and between these levels is a further source of complexity.
One of my favorite quotes about biology goes something like:
"An alien spaceship lands in your yard. You are kindly invited in to study and tour the ship. The technology is clearly billions of years more advanced than our own. Not decades more, not millennia more, not megayears more, but clearly gigayears more.
This is biology"
Sorry, can't find the exact quote and citation right now!