I am looking for an idiom or phrase that describes an unskilled person in a skilled job or about a person who is doing a job which is nowhere close to his professional skill.
He is ____________ in this job.
Such a person is in over their head:
be/get in over your head: to be or become involved in a situation in which you do not have the necessary skills, knowledge, or money to succeed
They are perhaps out of place (specifically in the context of the job).
He is out of place in this job.
out of place
not comfortable or suitable for a particular situation
He is worried about his job and feels out of place in a large organization.
Cambridge Dictionary of American Idioms Copyright © Cambridge University Press 2003. Reproduced with permission.
out of place
Not in the proper situation, not belonging; inappropriate for the circumstances or location.
For example, A high school graduate, she felt out of place among all these academics with advanced degrees , or This velvet sofa is out of place on the porch.
This idiom uses place in the sense of "a fitting position."
The American Heritage® Dictionary of Idioms by Christine Ammer. Copyright © 2003, 1997 by The Christine Ammer 1992 Trust. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Also, you can use the idiom "at home" with negation.
He is not at home in this job.
(Idioms) 27. e. at home,
well-informed; proficient: to be at home in the classics.
You can also say:
He is a misfit in this job.
Check the example phrase below.
3. a person who is not suited or is unable to adjust to the circumstances of his or her particular situation:
a misfit in one's job.
Inadequate to or unsuitable for a particular purpose.
- He is too incompetent to be trusted with such an important responsibility.
One of little consequence or ability.
- a political lightweight
phoning it in informal phrase
Work or perform in a perfunctory or unenthusiastic manner.
Idioms like "out of one's depth" cover the case where the relatively unskilled (or differently-skilled) person is not doing a good job in the more demanding position.
To address the case where the unskilled person is doing surprisingly well in the position, you can consider the idiom "punching above one's weight".
He had no formal financial education and no previous experience with the markets, but the senior brokers at Morgan Stanley were pleasantly surprised to see that he had found his feet very quickly, turning a tidy profit within a mere week of joining the firm. "The new boy's certainly punching above his weight", enthused the big boss at the weekly board meeting.
Such a person is a jack of all trades, master of none.
Jack of all trades, master of none: (figure of speech) "Often used in a negative light to describe someone who can do many things, but is not good at any one of them". (Goodenglish.org.sg)
EDIT: As per the OP's sentence with my answer in situ: Thus, "He is a jack of all trades, master of none in this job". I see the proverbial jack here as someone who posses certain skills but none coming close to what is required or needed for the job he is doing as per the OP's, "...or about a person who is doing a job which is nowhere close to his professional skill".