I believe “genteel” comes from the same root as “gentleman” (in the sense of social class, not in the sense of manners as we often use it in the US).
There is a line in Pride and Prejudice (at least one of the movies) near the end where Elizabeth says to Lady Catherine that there is no difference of rank between herself and Mr. Darcy:
“He is a gentleman, and I am the daughter of a gentleman.” Or something like that.
Mr. Darcy is both rich and related to nobility (Lady Catherine, for one), so it’s easy to see how he would be considered a gentleman. He has no title, so he is not actual nobility himself. I believe Lady Catherine married into nobility, so she was merely gentle before that.
Mr. Bennett (Elizabeth’s father) has some kind of fairly modest inherited income, so he doesn’t have to work for a living either. During his life, at least, he owns the source of the income, even if it passes away from his immediate family on his death. (I believe it is mentioned in the book that Mr. Bennett married beneath his class, and this is one explanation for why Mrs. Bennett is so vulgar. Presumably she had other charms at the time which have faded after the rigors of having 5 children.)
To look at another character from an old movie, in “The Old Dark House” Charles Laughton plays a very rich man who is not a gentleman. He is an industrialist of some kind instead.
So a gentleman appears to be someone whose income is based on what he owns, rather than on what he does (Charles Laughton’s character works very hard being an industrialist; Mr. Darcy probably has someone to manage his estates for him). There are, apparently, exceptions, for a gentleman, especially a younger son, may go into a profession like the law, the clergy or the army (as already mentioned). A gentleman may not go into Trade or get a Job working for someone else, on pain of becoming middle class (a step down from gentility).
All this is a long way around the barn to say what has already been said - shabby gentility means that you can be received as a social equal by other gentle people due to your ancestry, but you do not have the means to hobnob with them as much as you would like. You must pinch pennies. You (or your wife if you are a man) will make many or most of the family’s clothes, especially the women’s clothes (men’s tailoring is rather more difficult). You may owe money at some of the shops, but since you are a gentleman they don’t send the sheriff after you to pay, nor take you to court, as long as you make some effort to pay some of it. You may be constantly overdrawn at your bank (which just means that the bank is giving you short-term loans to cover your checks). If you are lucky and can get away with it, you will be going out to dinner at other people’s houses much more often than you have them to yours. Especially if you have relatives who are better off - it is permitted to be too proud to accept actual money from them, but not too proud to eat at their table frequently. Shabby gentility was often the lot of spinsters who had no close relations with whom they could live and off of whom they could sponge.
This was life as a shabby genteel person in 19th century England, at any rate.
Roddy