The leech, Roger Chillingworth does indeed function figuratively as described above. In another sense, Roger will not let go of this situation just like a leech gets stuck to host through the power of suction. He is bound and determined to get the two sinners to confess the name of the father of the child.
Chillingworth, having this great pursuit, takes residence in the town. Hawthorne points out that this is of great significance because of Chillingworth's abilities. Doctors of his capability often settled a lot closer to Boston wherein there would be a great body of people to serve as well as schools of medicine to take up instruction.
Likewise, Chillingworth is well on in years.
This learned stranger was exemplary, as regarded, at least, the outward forms of a religious life, and, early after his arrival, had chosen for his spiritual guide the Reverend Mr. Dimmsdale.
People understood him to be an experienced and intellectual man. He was an admired practitioner of faith. Yet among these great qualities, he chose a timid and young spiritual mentor. This is odd. He attached himself to this young man and must have a reason for it.
About this period, however, the health of Mr. Dimmsdale had evidently begun to fail.
Conveniently, the young reverend had very recently made the acquaintance of a leech. The ill in him is doing him more harm than good, just like the effort of leeches in those days to indeed suck out the 'bad blood.'
As you say, this word used to mean a doctor and so Chillingworth can be described using that word. However, the word also means a kind of slug like thing that sucks blood. Doctors were called this because they used to put leeches on people to suck their blood (back when they used to think bleeding was good for you).
The idea here is that Chillingworth is sort of sucking Dimmesdale's blood. Not literally, of course, but he is sort of sucking away Dimmesdale's energy and desire to live.
You can also argue that Chillingworth is actually pulling the "bad blood" out of Dimmesdale by pulling bad memories and guilt out of him.
Chillingsworth is a doctor, explaining the technical use of the word leech, but his personality justifies its connotations. Chillingsworth is a leech in that he feeds off of the lives of others, not able to rely only on himself. First he was a leech off of Hester, finding his happiness in her while she lived in an unfulfilled marraige. Then, he moves in his vengeful ways to Dimmesdale. Like a leech used to be believed to do, he is there to cure while actually causing harm. He is sucking something out of Dimmesdale, but it is not the guilt and shame that is poisoning him. Instead, Chillingsworth bleeds Dimmesdale dry of his will to live, secretly and under the pretence of a healer.
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The term leech was originally attributed to doctors in the olden days because they used leeches to treat many different things especially infection. However, Roger is a leech in the sense that he is grabbing onto the Reverend because he wants to suck the secret out of him.
Dimmsdale can not escape his grasp as he has latched onto him mentally and physically as his physician and supposed friend. He waits with baited breath until he has the chance to look upon the Reverend's chest and learn his secret.