Are the dual transportation and learning meanings of both "coach" and "train" just a coincidence?

In a learning context, you have one individual who "coaches" and another who "trains".

In a transportation context, "coaches" and "trains" are both methods of transport.

Is this just a coincidence or is there some root to both words which can explain a link between transportation and learning?

Yes, the meanings are related.

Train was first applied to the modern vehicle in the sense of a "train of wagons." Train was used (and still is) to talk about making branches or vines grow along a certain path. The idea with "train" someone is to help them get on a path and stay on it. It seems that it was mainly used for physical things such as training a horse, or training soldiers. I was not able to locate more examples but I would think that "train" might also have been used for swordsmanship and also musical activities like playing the violin or singing. We also speak of a "train of thought."

"to discipline, teach, bring to a desired state by means of instruction," 1540s, probably from earlier sense of "draw out and manipulate in order to bring to a desired form" (late 14c.), specifically of the growth of branches, vines, etc. from mid-15c.
_Etymonline

Coach was a large kind of carriage. It was natural to transfer this to a train (the transportation vehicle). Coach can also mean to carry someone in a coach. The first recorded use of "coach" as a verb in the modern sense of "help someone with an activity" was in 1849.

1610s, "to convey in a coach," Meaning "to prepare (someone) for an exam" is from 1849
_Etymonline

As a side note, this "backward looking" definition of "coach" is useful because it helps distinguish the difference between a trainer and a coach, a coach being a person who "carries you along" and gives you support and guidance. This sense is used in the modern expression "life coach" as someone who supports you, helps you through challenges or goals, and in general gives you a "push."

I believe the two terms are only indirectly related, as train in this sense does not come from the mode of transportation at all. Both do come from the larger concept of providing direction.

The term train has long roots (pun intended) in agriculture, and means to direct or correct, and was primarily used to keep vine plants from overgrowing into neighboring vineyards or out of fertile soil / good sunspots, etc. This term then became the term for supply lines in an army (commonly called "the baggage train" in the literature), and this is where the verb to train in the sense of learning comes from - the act of teaching horses, cattle, elephants, etc. to stay in line and the act of teaching men how to correctly perform their duties in the train.

So rather than transportation, the verb comes more from pathfinding - hence routine, orientation, etc.

Whereas coach as a term does derive directly from coaches.

1. Coaching used to mean the act of using a coach to get from place to place. But coaching also refers directly to the coachman's job directing the horses.
2. "Coaching up" meant to coach uphill, which was a particularly arduous task and horses would often simply stop at the first sign of effort, requiring the coachman to whip the horses and basically force them to finish climbing the hill.
3. Then the term "coaching up" became slang that meant, variously, to "learn" ("coaching up the various subjects of public interest in Eccleston"), to "produce through forced or extra effort" ("all attempts at coaching up a sensation proved a failure"), to "study" ("every Civil Service student who has been coaching up his history will recollect ..."), and "to produce material for learning" (an hour in the evening occupied in "coaching up" work ... - note the quotation marks).