Bruising is common after having blood taken.
The part that seems the most worrying is that it's spreading.
Now here's a guess: if it were a bacteria, then there could/should be a bump and/or pus where the stab went in. So maybe it's a virus. Does it look like this photo with the bumps and redness:
where you scroll down to 'Herpetic whitlow'?
"It looks almost like a burn on my fingertip"
No swelling or heat then? But just redness, and some tenderness... so it seems to be mild (for now) inflammation.
Is it possible you are having a reaction to whatever was used to clean the skin? But then again the timing is probably wrong for that, being weeks later.
An ER nurse would probably know in an instant what it is. Sorry, but I myself don't know much further. It would be easy to say 'go to an ER or a walkin clinic' -- but they're not free, of course.
Btw, with that photo I'd meant that you might have just an early stage, beginning version of that type of virus infection. It does take weeks to develop.
You might take a picture and post that on your personal page here.