Oh dear, this seems to be several questions, I hope that's okay as they're all the same novel. Spoilers, if you haven't read it.(Also, here's the book on Gutenberg). Breaking that down:
Question 1: Obviously, it's a fiction novel, but Jules Verne was well learned and I'm wondering if these folks are prototypes of folks that might exist. German professors obviously exist, but would they know everything Vernes' professor knows?
The professor is introduced as:
My uncle was a German, having married my mother's sister, an Englishwoman. ...This home was in a large town, and my uncle a professor of philosophy, chemistry, geology, mineralogy, and many other ologies.
That's a lot of ologies, so straight up is it possible for someone to even have those qualifications? He seems to be around 40, which I base on his ~20yo nephew and his otherwise very good physical health.
The tools he brings are:
A centigrade thermometer of Eigel
A manometer worked by compressed air, an instrument used to ascertain the upper atmospheric pressure on the level of the ocean. Perhaps a common barometer would not have done as well, the atmospheric pressure being likely to increase in proportion as we descended below the surface of the earth.
A first-class chronometer made by Boissonnas, of Geneva.
Two compasses, one for horizontal guidance, the other to ascertain the dip.
A night glass.
Lastly, two Ruhmkorff coils, which, by means of a current of electricity, would ensure us a very excellent, easily carried, and certain means of obtaining light and a voltaic battery.
Some examples of his feats:
Feat 1: Speaks several languages - Latin, Italian, French, Icelandic, Danish, English, German.
Feat 2: Judges by sight of the rocks around him what era of time they're in (e.g. when they're coming out of the volcano at the end), or when they're descending into the earth at the beginning.
"We are still within the region of the primordial soil, the soil in which took place the chemical operation of metals becoming inflamed by coming in contact with the air and water....This is gneiss. This is mica schist--siliceous mineral...this is the epoch of transition,..."
Feat 3: Identifies a Ichthyosaurus and plesiosaurus by sight.
Feat 4: Picks up a bunch of bones and be able to identify them not just as a fish, but of a specific type of extinct fish:
The Professor examined the fish carefully, noting every characteristic; and he did not coincide in my opinion. The fish had a flat head, round body, and the lower extremities covered with bony scales; its mouth was wholly without teeth, the pectoral fins, which were highly developed, sprouted direct from the body, which properly speaking had no tail. The animal certainly belonged to the order in which naturalists class the sturgeon, but it differed from that fish in many essential particulars.
"This fish, my dear boy, belongs to a family which has been extinct for ages, and of which no trace has ever been found on earth, except fossil remains in the Devonian strata. To the order of Ganoides--an order of fish having angular scales, covered with bright enamel--forming one of the family of the Cephalaspides, of the genus Pterychtis.
"a man of the Quaternary period of our globe....I dare venture to say that this human sample or specimen belongs to the Japhetic family, which spread over the world from India to the uttermost limits of western Europe.... The facial angle is formed by two planes--one more or less vertical which is in a straight line with the forehead and the incisors; the other, horizontal, which passes through the organs of hearing, and the lower nasal bone..."
Overall Questions:
Is this at all realistic? Would such a learned man exist in 1863, and if so, could he feasibly do all of these things?
And that leads me to questions 2 and 3 - they just sort of set off for four months and come back and everything is jolly. At what point would they have been declared dead? If so, would the estate have been auctioned off? Was it common to make multi-month trips?
Edit: I just learned there is an eponymous movie. I have not seen it; my questions are based around the novel.
You must remember that a century or more ago, it was possible to read virtually everything in any given subject, and given the strict nature of formal education, proficiency in numerous languages was required. In addition, advanced education was often the luxury of the affluent who had sufficient financial resources to indulge in all sorts of expeditions and intellectual digressions.
My mentor, Sven S. Lilejblad (1899-2000) was born later than your character, but whatever he was able to achieve in the early twentieth century - regarding diversity of fields, etc. - was even more available to a scholar in 1863. Sven's first degree (1924) was in "social studies": he was required to study in all the fields of what we would call the humanities and social studies. He participated in an excavation of a megalithic tomb; he studied psychology under a student of Freud; he studied sociology, economics, and ethnography; and he studied history. He was also required to know Greek, Latin, German, French, and English. He had already become proficient in Russian and Finnish for various reasons, and he could read and speak Danish, Norwegian and Old Norse, besides his native Swedish. He wrote his dissertation in 1927 and had published in six languages by the time he was thirty years old.
Sven was thrown into financial destitution when he prepared to accompany the great Swedish explorer, Sven Hedin (1865-1952), who had invited diverse young scholars, including Liljeblad, to go on his final expedition to Asia in 1927. With borrowed money, Liljeblad purchased equipment for himself and a friend, who would repay half the loan. The friend sold everything and disappeared, leaving Liljeblad destitute and unable to undertake the journey. As a result of this, Sven was forced to work to repay the debt, and he undertook the writing of dissertations for wealthy students who wanted PhDs but had no interest in doing the work (or learning anything). Sven's body of research was consequently extremely diverse as subjects undertaken were in all the fields of the social sciences but even beyond that. (For some of the background here, see my brief article, Nazis, Trolls and the Grateful Dead; in addition, the Special Collections Library at the University of Nevada, Reno has Sven's papers, including excerpts from my body of letters from Sven and from one of his students from the 1930s.)
Having known Sven, it is easy to imagine the German professor described by Verne. These sorts were not common, but they were not outside of the realm of possibilities. Education pointed one in the direction of being a "Renaissance Man" (given the sexism of the time - but there were also plenty of women with diverse interests and expertise!). Obviously some people were able to rise to the occasion more than others, but given the possibility that a world with limited bibliography presented, it was to write and speak with authority on a number of subjects.
To that final point, I will point out that while working in the university library in 1975, I listened as the staff in Bibliographical Search Department described how a recently published article had asserted that in that year there were to be as many books published as in all the years previously - combined! The article also asserted that in 1976, the publishing industry anticipated that the same would be true - more books in 1976 than in all the combined publications in human history. I did not read the article, and I cannot attest to its accuracy, but the mere fact that the librarians were discussing the tidal wave of publications that was occurring in the 1970s (and has continued during every decade since) helps put the earlier, scholastically innocent time of the nineteenth century into perspective.