Can anyone identify the uniform in this portrait?
Not easily, there isn't enough detail.
According to Wiklipedia:
The entire Danish Army wore red coats up to 1848 and particular units in the German, French, Austro-Hungarian, Russian, Bulgarian and Romanian armies retained red uniforms until 1914 or later. Amongst other diverse examples, Spanish hussars, Japanese Navy and United States Marine Corps bandsmen, and Serbian generals had red tunics as part of their gala or court dress during this period. In 1827 United States Artillery company musicians were wearing red coats as a reversal of their branch facing colour. However the extensive use of this colour by British, Indian and other Imperial soldiers over a period of nearly three hundred years made red uniform a veritable icon of the British Empire.
You might ask on Bruce Bassett-Powell's Uniformology