...then, tying the horse to the hitching rail...
What is the function of "hitching" here?
I initially thought it was an adjective, but could not find a translation in any dictionary indicating that it is an adjective.
You are correct that hitching functions in your sentence as an adjective. You don't find it in the dictionary in that form, however, because it is not actually a separate word. It is the present participle of hitch* acting as a verbal adjective, or participial adjective. (Although see below.**)
Typically, multiple forms of verbs are not listed in dictionaries; only the bare infinitive is given an entry, while other forms may be listed under that entry (especially for irregular verbs). This is true even though some of those other forms can be used as other parts of speech (like in this question).
There is a gray area, however, in that some words that start out as participles may become separate words in their own right, through heavy use, or because they gain an additional figurative meaning. Some examples are interesting, sleeping, and killing. The exact line at which a participle "graduates" to become a separate word is, I imagine, one of those areas without hard and fast rules. For instance, the OED, while not giving hitching a separate entry, does list it as a "derivative" of hitch.
(In fact, the OED lists both a noun and adjective derivative for hitching. It further implies that formations like the one in this question (and hitching post, etc.) actually derive from hitching as a noun. So by that logic, hitching in hitching rail is actually an attributive noun, which has graduated from a verbal noun. An attributive verbal noun, if you will. To the extent that there is a distinction between that pathway and the verbal adjective, I can't quite wrap my head around what it might be.)
* Hitch in this context refers to tethering a horse, and so a hitching rail is a rail to which horses are commonly tethered.
** EDIT: Having given it a little more thought, the pathway from verb to adjective via noun may make more sense. I think the present participle adjective usually (perhaps always) correctly modifies a noun that would be the subject of the verb. An interesting book is a book that interests. If a cat sleeps, it is a sleeping cat. Rail is neither the subject nor the direct object of hitching (the horse is the object). We would think of a hitching rail neither as a rail that hitches nor as a rail that is hitched, but as a rail for hitching ("The rider hitched the horse to the rail."). Hitching in the for phrase must be a verbal noun, and so it is the noun form that is used as the adjective. If the noun is the object of the verb used as an adjective, then the past participle is used: is someone plows a field, it is now a plowed field, and a horse that has been hitched is a hitched horse.
If you had left your horse in the care of a saloon employee who is involved in tethering everyone's horses to a rail, you might say that "the hitching man has hitched my horse to the hitching rail". This is either uselessly confusing or illustrates the difference between the use of hitching as a present participial adjective ("the man is hitching") and the attributive verbal noun ("the rail for hitching"). There is some brief discussion of this at the wikipedia page on attributive verbs (see the "swimming competition" part).
Ultimately, these technicalities don't affect the main answer to the question: Yes, hitching is functioning as an adjective in this context, but it is not in the dictionary because it is a verb form.
Yes, "hitching" here is an adjective.
To "hitch" is to tie two things together. I can only think of four cases where the word is commonly used:
(a) Hitching an animal to a post or rail. That is, tying the animal to a fixed object so it doesn't wander off or run away. When people commonly traveled on horses, public buildings often had "hitching posts" or "hitching rails" to tie the animal to. https://www.flickr.com/photos/lakelife1/4093723938/?ytcheck=1 It is a "hitching rail" because it is a rail used for the purpose of hitching, i.e. hitching horses.
(b) An informal term for marriage. "Bob and Sally is gettin' hitched next Sunday."
(c) Hitching an animal to cart or wagon. Tying the animal to the vehicle so the animal can pull it.
(d) Metaphorically from (c), "hitched his wagon to a star", meaning attaching your life to some person or organization that you expect to be successful so that you can benefit from their success. "Sally hitched her wagon to a star when she became Bob's personal assistant. His career really took off and she followed him all the way." (And I am suddenly reminded of a lecture I heard years ago by a man who became the target of numerous lawsuits and went bankrupt. At one point he said that his wife "hitched her wagon to an anvil".)
Verbs can be turned into gerunds.
To hitch -> hitching.
The rail is for hitching.
Then the gerund, like any common noun, can be used as a adjective:
Tie the horse to the hitching post.
To color -> coloring.
The child was given a coloring book and some crayons.
To shuck -> shucking.
At the raw bar, the restaurant worker was shucking oysters with a shucking knife.
To iron -> ironing.
We keep the ironing board in this closet.
to birth -> birthing
The mare was moved to the birthing stable.