Is it
An Intel x86 processor
Born in the 70s, I'm everywhere today.
The first x86 chip, the 8086, was introduced in 1978.
I start as real but get in people's way.
The 286 architecture introduced protected mode, but they still booted in real mode for compatibility reasons.
After generations, I still remember my roots.
Each new processor design was referred to as a generation, but they remain backward compatible with the earlier generations
Look closely, you'll find biology in my boots.
The computer boots from a BIOS, which contains the word Bio
When I'm protected I perform at my best.
Protected mode enables features like virtual memory and paging, which are pretty important in a modern computer
You'll see me with flags after a test.
The FLAGS register is the status register in Intel chips, and the TEST assembly instruction results in 3 flags being modified.
For purists I offer no charm.
A purist might prefer a RISC architecture
I have an eye but no arm.
Intel begins with an I, and ARM are a competitor.