OK, there's a bit to unpack here. Forgive me geographical shift, because my main library regards daily life in XIX century Poland and Russia but most of the sentiments were widespread enough to be discussed internationally.

Firstly- Ladies did travel and knowledge of the languages were a normal thing in that part of society to the point that aristocratic Polish and Russian families were more fluent in French and English than their native languages. People of means traveled for fun, health, shopping, politics, networking. Moreover French was a lingua franca of literature and culture- literature in languages regarded as vernacular was slowly and painfully emerging in first half of the century, translation was more a freestyle pastime- in surviving libraries from XVIII century you will certainly find more French literature than everything.
In XIX century travel and literature tickled down to emerging middle class- reading diaries you can see that at least summer in the countryside was seen as necessity for physical and mental health with international travels to good resort and famous places being a good step up.
Embroidery and needlework was a symbol of femininity for time immemorial.
In Roman texts virtuous wives work with wool. When in XVI century Sigmundus the Old presented his daughters to Russian envoys they "just happened" to work with cloth. Up to XX century in pastime was praised as way to avoid idleness.

No wonder that those "accomplishments" ended into curriculum of emerging middle class.
Having a governess or sending daughter to private school instead of having them help in family business was a status symbol as much as stay at home wife totally not preoccupied by anything but domesticity.
Ads put in newspapers by potential governess give us a glimpse of what was expected- arithmetic, history, botany, music, French and English. Needlework was absent but so was cooking- it was expected that daughters would be taught by their mothers or female guardians.

The quality of that education varied to put it mildly- there was little to no control over private education. Educators raised the concern that parents often could not check the level of private tutors or just choose the cheapest ones and governesses who could prove their qualifications with diplomas charged extra. That lead to glut of young woman with very spotty knowledge about anything and phases learned by heart passing as language fluency.

It must be said that such "salon education" was under heavy criticism- the same educators rightfully said that even very accomplished young woman is left with no marketable skills, rendering her helpless in any kind of crisis. Their answer was a rise of new type of the school know as a Governess Institution that- apart from good language, history and literature classes incorporated elements of hygiene, gymnastic and modern home economics with goal of raining independent young woman.

Most of my primary knowledge comes from
Życie Codzienne Polaków w XIX wieku (Tomy I-VIII) Uniwersytet Łódzki

https://czasemancypantek.pl/w-spoleczenstwie/edukacja (primary sources on life of Polish women from XIX century)

Knowing French would be your ticket to "exciting exchanges with foreigners" in this period. As others have pointed out, French was the lingua franca of the European upper class in this period, much like English is for a global upper class today. It was considered a necessary accomplishment for educated gentlemen too, but they often had less time to study it and perfect it, because they either had to learn Latin and Greek (if they were being educated for a career in the church, in science or medicine and/or at a public school) or were too busy with more practical business, military drills, money-making etc.

​

If a young woman hoped to marry well and become the mistress of a household, preferably in London, which entertained distinguished foreign guests (royalty, nobles, diplomats, business tycoons, artists, scientists, various celebrity travelers etc.) she would be well advised to learn French well, as she then would be able to communicate with these guests and make them feel welcome, as a perfect hostess. She would also be able to act as interpreter for her husband or other Englishmen who didn't know French as well and could thus play a role in politics and business dealings and more cultural exchanges. The same applies of course to letter writing in French, the preferred mode of international communications, and to women serving as diplomatic hostesses abroad as wives of diplomats. And women as courtiers to foreign-born royalty who didn't know English well.

​

A good example is George Washington, who was somewhat handicapped as an international figure, leader and celebrity by his lack of French skills. I don't know whether his wife knew French and helped him out, but consider how close a woman could get to important personalities like Washington and Lafayette (or other foreign envoys) if she could translate between them. (Though Lafayette apparently quickly learned English.) Most upper class foreigners, whether French, Spanish, Italian, German, Dutch, Scandinavian, Polish or Russian did not learn English in the 18th century. English was seen as an obscure language fit for sailors and merchants and their coarse business dealings. It was not until Britain appeared as a major imperial and industrial power in the Victorian age in the 19th century that there suddenly was a craze for English nannies and governesses and many upper-class children all over Europe grew up learning English in their nurseries.

​

And perhaps the English prided themselves a little extra on genteel French skills because of the Norman French origin of their aristocracy and the massive amounts of French loanwords in English? (In countries such as Russia or Denmark speaking French was seen as entering into a more civilized and refined space. I don't get the feeling the English viewed it as such. It was just another aspect of being genteel - English gentlefolks.)

​

Source: College history subjects, extensive reading, also about linguistics and participation in the Alexander Palace Forum about Russian and European Royalty History: https://forum.alexanderpalace.org

There's always more that can be said, but I have a previous answer on this general subject that I'll paste below:

In the eighteenth and nineteenth centuries, it was very important for young women of the middle and upper classes to be "accomplished", in the parlance of the day. Since you've read Pride and Prejudice, you probably remember this:

“It is amazing to me,” said Bingley, “how young ladies have patience to be so very accomplished as they all are.”

“All young ladies accomplished! My dear Charles, what do you mean?”

“Yes, all of them, I think. They all paint tables, cover screens, and net purses. I scarcely know any one who cannot do all this, and I am sure I never heard a young lady spoken of for the first time without being informed that she was very accomplished.”

“Your list of the common extent of accomplishments,” said Darcy, “has too much truth. The word is applied to many a woman who deserves it no otherwise than by netting a purse, or covering a screen. But I am very far from agreeing with you in your estimation of ladies in general. I cannot boast of knowing more than half a dozen, in the whole range of my acquaintance, that are really accomplished.”

“Nor I, I am sure,” said Miss Bingley.

“Then,” observed Elizabeth, “you must comprehend a great deal in your idea of an accomplished woman.”

“Yes, I do comprehend a great deal in it.”

“Oh! certainly,” cried his faithful assistant, “no one can be really esteemed accomplished who does not greatly surpass what is usually met with. A woman must have a thorough knowledge of music, singing, drawing, dancing, all the modern languages, to deserve the word; and besides all this, she must possess a certain something in her air and manner of walking, the tone of her voice, her address and expressions, or the word will be but half deserved.”

“All this she must possess,” added Darcy, “and to all this she must yet add something more substantial, in the improvement of her mind by extensive reading.”

“I am no longer surprised at your knowing only six accomplished women. I rather wonder now at your knowing any.”

"Accomplished" and "accomplishments" aren't just random word choices on Austen's part - they're what this society used to refer to women's non-essential, non-"work" skills. Being able to dust a room wasn't accomplishment and making jelly or preserves wasn't an accomplishment, because these were things that servants and housekeepers did, as well as women who couldn't afford servants, and in turn, servants, housekeepers, and women who couldn't afford them didn't have time to practice accomplishments. Miss Bingley's list of music (playing the piano or harp), singing, drawing (which was probably meant to encompass watercolor painting as well), dancing, and speaking/reading other modern languages (French, German, and/or Italian) states the typical list of necessary accomplishments for an upper-class woman, although it neglects embroidery. Women would learn these skills as children and teens, either at home or at a boarding school. If they learned at home, the skills would be taught to them by a governess, maybe with the help (if their parents could afford it, and/or very concerned with accomplishments) of an outside male teacher in music/dancing or art.

The concept was much critiqued at the time by activists for women's rights. Basically, young men of the same social status were educated to do white collar work, becoming a clerk, lawyer, doctor, etc., or to go on to university and then be landowners, while their sisters and future spouses learned these "ornamental" skills which could only be used as a source of income with a significant step down in social status and most likely material comfort. This definitely wasn't wrong. However, modern people taking this critique at face value has led to the general neglect and underestimation of the use value of accomplishments. They can also be seen as job skills in their own right, for the position of upper- or middle-class wife - skills that young women could work on to improve their chances on the marriage market. Far from being mindless and stultifying, these skills require a lot of practice to achieve mastery or even proficiency, and could be very useful for entertaining oneself and others, communicating, and adorning one's space.

ANYWAY. I'm getting around to your question now. The point of accomplishments was in the breadth, not the depth - the goal was to be proficient in several rather than a master of one specifically, which could imply some kind of vocational training, or the lack of enough money to pay for the proper instruction (given the class-based nature of the game). The actual paintings themselves, or the embroidered pictures or samplers, or the sung songs and played music, were also not the point. While a typical accomplished woman's family might value her paintings and visitors might comment favorably on them, she was not going to professionalize her talent unless she were in dire straits, so she would not become known as a painter and therefore future generations would have little reason to remember their names and pass them down along with the paintings, if they even did pass down the paintings at all, and so they didn't achieve any kind of fame and have not been written about. When these artworks still exist, they're generally in attics and local historical societies, and may be more known for being the product of a particular school than for their specific maker.