The answer is yes, the color obtained from Alizarin after mordanting with Aluminium metal salts is also called Turkey Red. This is the red color popularly used in Dabu, Bagru, Tarapur, Kalamkari and Ajrak handblock prining of India. Alizarin is a mordant dye. Mordant is an additive which improves the ability of a dye to bond to the fabric.
Mordant dyes are used in combination with salts of metal ions, typically aluminum, chromium, iron, and tin. The metal ions adhere to the fabric and serve as points of attachment for the dye molecules.So Aluminium present in Alum is used to generate that famous red color of alizarin. Other metal salts can be used as mordants for alizarin with some interesting color changes results. For example iron sulfate as the mordant imparts a rich brown color to the dyed fabric.
Alizarin, also spelled Alizarine, a red dye originally obtained from the root of the common madder plant, Rubia tinctorum, in which it occurs combined with the sugars xylose and glucose. The cultivation of madder and the use of its ground root for dyeing by the complicated Turkey red process were known in ancient India, Persia, and Egypt; the use spread to Asia Minor about the 10th century and was introduced into Europe in the 13th.
Laboratory methods of preparing alizarin from anthraquinone were discovered in 1868, and, upon commercial introduction of the synthetic dye in 1871, the natural product disappeared from the market for textile dyes.
There is a difference in Turkey Red and Turkey Red Oil. Turkey red oil, also called sulphonated (or sulfated) castor oil, is made by adding sulfuric acid to vegetable oils, most notably castor oil.It was the first synthetic detergent after ordinary soap. It is used in formulating lubricants, softeners, and dyeing assistants.
A trivia- Alizarin is something of a fungicide and pediculoside (that is why it found in the soldiers uniforms).
Sources:
https://www.academia.edu/24482692/Natural_Dyeing_Fabric_dyeing_with_Madder
https://en.wikipedia.org/wiki/Alizarin