Although it is against the prevailing opinion here so far, I believe that “she has my hand and arm” does refer to the attractiveness of Frederica’s hand and arm. Lady Susan believes that if Frederica can learn to play at least with “taste” and “assurance”, even if not with talent, that will be enough to allow her to attract potential suitors by showing off her physical attributes at the piano.

The following sentence makes it clear that it cannot mean “has my talent in playing”, as Lady Susan has no talent in playing: I was so much indulged in my infant years that I was never obliged to attend to anything, and consequently am without the accomplishments which are now necessary to finish a pretty woman.

The text continues:

Not that I am an advocate for the prevailing fashion of acquiring a perfect knowledge of all languages, arts, and sciences. It is throwing time away to be mistress of French, Italian, and German: music, singing, and drawing, &c., will gain a woman some applause, but will not add one lover to her list - grace and manner, after all, are of the greatest importance. I do not mean, therefore, that Frederica's acquirements should be more than superficial, and I flatter myself that she will not remain long enough at school to understand anything thoroughly. I hope to see her the wife of Sir James within a twelvemonth.

Support can be found for this interpretation in explanatory notes at the end of some editions of the novel or in some of the literary criticism on the novel.

From Lady Susan, the Watsons, and Sanditon: Unfinished Fictions and Other Writings
Explanatory Notes
I want her to play ... my hand & arm : viewed cynically, as here, for a young woman on the marriage market her performance at the piano or harp were as much an opportunity to show off her physical assets as her musical abilities.
https://books.google.co.uk/books?id=L7wyEAAAQBAJ&pg=PA251&lp...

From The Routledge Companion to Jane Austen
Chapter 17 Jane Austen and Music
In addition to demonstrating her obliging nature, modesty, and musical skill, the performer also displays her body to the male gaze. The title character of Lady Susan, though sceptical of the range and depth of accomplishments expected of a genteel young woman, understands this benefit of music education. Lady Susan reasons that ‘Grace and Manner after all are of the greatest importance’ and more likely to add ‘Lover[s] to her list’ than the more academic subjects of a girl’s education (LS 199). Therefore, she wants her daughter ‘to play and sing with some portion of Taste, and a good deal of assurance, as she has my hand and arm’ (LS 199). Since Lady Susan never learned to play, it is Frederica’s physical resemblance to her that she wants potential suitors to notice. Frederica is too meek to exert much agency in courtship, but Lady Susan demonstrates her ability to attract and manipulate men throughout the novella and is alive to the seductive possibilities of music performance. Thus, despite piano method books’ attempts to depersonalise a woman’s performance, Austen notes that a female musician draws notice, at the very least, to her hand and arm.
https://books.google.co.uk/books?id=3ddCEAAAQBAJ&pg=PA223&dq...

An interesting blog on the theme

On the subject of playing musical instruments it is clear that no opportunity must be lost to display the performer in the best possible light.
“Let their attitude at the piano, or the harp, be easy and graceful. I strongly exhort them to avoid a stiff, awkward, elbowing position at either; but they must observe an elegant flow of figure at both.”
Playing an instrument and singing were basic accomplishments for any young lady and she was expected to help provide the entertainment at family gatherings and social occasions. Not only was this (hopefully) pleasant for the listeners, but it demonstrated her taste and allowed her to be viewed at her best by potential suitors. The ‘Lady of Distinction’ makes this display function exceedingly clear. She considered the harp showed “a fine figure to advantage. The contour of the whole form, the turn and polish of a beautiful hand and arm, the richly-slippered and well-made foot on the pedal stops, the gentle motion of a lovely neck, and above all, the sweetly-tempered expression of an intelligent countenance; these are shown at a glance, when the fair performer is seated unaffectedly, yet gracefully, at the harp.”
https://janeaustenslondon.com/2021/02/24/happily-adapted-to-...