I'm not all that up to snuff on classical music history but as far as I know the last time we saw a child play like Alexandra Dovgan was Clara Schuman. Does anyone else know any other child performers since the time of Clara Schuman?
David Bongartz (better known by his stage name "David Garrett") started at age four and was in concert with the Hamburg philharmonic orchestra by age ten (I vaguely seem to remember hearing him with the Bach violin concerto #1 in A minor in Aachen at about that time).
Note that it is not actually unusual for children to start on the violin at rather young age and a considerable showcase element particularly for a monophonic instrument like the violin is raw virtuosity that will not depend as much on refined and matured musicality for a compelling performance as may be the case with some other instruments: you don't hear much of harmonica prodigies or recorder prodigies, even though adult players like Toots Thielemans or Frans Brüggen have cut out niches there.
That, in addition to the availability of viable smaller-size instruments, may be one reason that violin prodigies are comparatively more frequent in public than prodigies on other instruments.
Piano, for example, is comparatively unfriendly as an instrument for hands of small children when played at an advanced level.
In the world of violinists it is almost normal for virtuosi to start at a very young age. However if you are asking specifically about pianists then the obvious answer is Evgeny Kissin.
According to Wikipedia, Clara Schumann started receiving basic piano instruction from her mother at the age of four and made her official debut at the Gewandhaus in Leipzig, at age nine.
Again according to Wikipedia, Evgeny Kissin began piano studies at the age of six at the Gnessin Music School in Moscow and made his debut at the age of ten, performing Mozart's Piano Concerto No. 20 in D minor with the Ulyanovsk Symphony Orchestra.
According to Kissin he started started singing in his cot at age 11 months the theme of the Bach fugue his sister was learning on the piano at the time and to play the piano as soon as he could reach up to the keyboard at the age of two.
If the question is about child prodigies, check violinist Salvatore Accardo.
At age three he got a violin, took it, and started playing right away.
Here's one of the interviews in which he tells that story. He also tells about discussing the issue of child prodigies with other musicians, and discovering that it is not uncommon.
(Links to the original Italian and with google auto-translate)
This phenomenon is far rarer than may be thought. Alexandra immediately put me in mind of Josef Hofmann and Solomon. The key is in what Grigori Sokolov said of Alexandra: she speaks of music and plays it as if she were fully adult, yet without even a hint of precocity, which is often an unpleasant aspect of the so-called prodigy. It was this quality in Solomon that so delighted musicians of that time: Sir Henry Wood, Artur Rubinstein, Hamilton Harty, et al.