How is it even possible that Martha became the Doctor's companion?

In Doctor Who - Season 2 Episode 12 a character who looks identical to Martha is turned into a pseudo-cyber woman when a couple of ear pieces are fused to her brain. Later in the episode the ear piece is pulled out and a piece of what looks like her brain is clearly visible. It's stated that they can't be pulled out or that she'll die.

Then, just a few episodes later, she's a hospital worker and becomes the doctor's companion!

Is there any explanation for this?

We've already heard the doctor state that once someone dies it becomes a fixed event.

Answer

The same actress played two roles.

It's happened before, repeatedly (the Sixth Doctor's actor had a fairly major role as a secondary character in Five's time well before he was ever cast as the Doctor), and it will continue to happen (the Twelfth Doctor's actor played a Pompeiian during Ten's run).

Sometimes they make meta jokes about it, like Ten's observation that a major modern-era character in Torchwood looks exactly like the totally different 1800s-era character her actress played during Nine's run--or with Martha's reference to her cousin at Canary Wharf.

When a Time Lord companion of the Doctor was re-cast as an actress who'd previously played a character that the Time Lord had already met while being played the original actress, they called it out and had the character deliberately choose to regenerate into a form that mimicked the person she'd met. Similarly Twelve subconsciously modelled his new body on someone from his past.

Sometimes they ignore it entirely, as in the case of Colin Baker's stint as Commander Maxill or the fact that Amy Pond looks exactly like a Pompeiian soothsayer.

It's just an artifact of casting a long-running show that may demand dozens of extras for any given story. You double up, on purpose or by accident, because you liked the actor or because they're the best for the parts or because you can't find anyone else.