Learn English – Origin of “I’m gonna tan your hide”

The reference is to the tanning process in leather making (as you've guessed) and The American Heritage Dictionary of Idioms, Second Edition (by Christine Ammer) provides more details:

This term uses hide in the sense of "skin". The allusion in the first expression is to a spanking that will change one's skin just as chemicals tan animal hide (convert it into leather). [Second half of 1600s]

In ancient times, in the tanning process, the skin might have been beaten to remove the flesh and fat; however it is not the only method mentioned as some sources say that sharp stones or bones were used for scraping the skin as well. In the seventeenth century (the era when the idiom is originated), the tanning process didn't involve beating in the defleshing step. The skin is stretched and scraped with defleshing tools. Beating is mentioned in some sources as a method of softening the skin but it is usually done after the tanning process. (Here is an article about tanning in the seventeeth century: https://www.nps.gov/jame/learn/historyculture/tanning-in-the-seventeenth-century.htm)

Further supporting evidence from the book The Dictionary of Clichés: A Word Lover's Guide to 4,000 Overused Phrases and Almost-Pleasing Platitudes by Christine Ammer again:

tan someone's hide, to To give someone a beating. This term, in which the human skin is referred to as a hide (as it was from about the seventeenth century), may be on its way out, viewed with the same disfavor now accorded to _{SPARE THE ROD}. Nevertheless, during the years when corporal punishment was considered a normal procedure, it became a cliché. (Incidentally, the tanning process, in which animal hide is converted into leather, does not involve beating but rather a soaking in chemicals.) The expression dates from the seventeenth century. Charles Coffey used it in The Devil to Pay (1731): "Come and spin . . . or I'll tan your hide for you."

In OED, the earliest example is from c1670:

Expost. Let. Men Buckhm. 2/2     Let not your Worships thick skin be too sensible that we thus Tan your Hide.

The discoloration of the skin in the allusion can be explained with the use of tannin in the traditional tanning process which gives the distinctive color of the tanning. Here is an explanation from the book How it Works: Science and Technology, Volume 9:

Traditional tanning methods, which impart a characteristic color to the skins and convert an otherwise perishable material into a stable and nonperishable hide, traditionally used vegetable tannins leached from the leaves of shrubs and trees, such as quebracho and sumac leaves, and the bark of oak to tan the skins. Modern processes use concentrated vegetable extracts instead. The process was discovered early in human history, probably when a raw skin was left soaking in a pool of water into which tannins had coincidentally leached from fallen leaves.

Vegetable tanning produces the tan color still seen in sole leather and some harness and industrial leathers, such as belting. Apart from these applications, vegetable tanning is not widely used as the principal tanning agent, except in countries such as India. The process is slow: heavy sole leather, for example, can take several weeks to tan as the hides move progressively through tan liquors of increasing strength.

The discovery of the chrome tanning process early in the 20th century revolutionized leather making by introducing a mineral tanning process that can tan leathers in hours. Most shoe upper leather is now tanned in this way, and the process can be recognized by the blue-green color of the undyed leather. Chrome-tanned leathers were found to be more heat resistant and easier to waterproof and, in some cases, could even be made fully washable.

The flow diagram of the traditional tanning process from Davarg leather company: