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.