I've often encountered records of notable 19th-century figures being fluent in multiple different languages, often with two or three living languages and a couple of dead languages (Latin, Ancient Greek, etc.). A few examples:
Carl Friedrich Gauss understood most of the principal European languages, and "at the age of 62 he started learning Russian and in less than two years wrote and spoke it."
Chester A. Arthur conversed in Latin and Greek with other people who knew the languages.
Thomas Jefferson spoke English, French, Latin, and Italian, and claimed to be able to read Greek and Spanish. He also studied and wrote about Old English and German.
Helmuth von Moltke the Elder spoke and wrote German, Danish, French, English, Italian and Turkish.
Pedro II, Emperor of Brazil was claimed to speak and write in Portuguese, Latin, French, German, English, Italian, Spanish, Greek, Arabic, Hebrew, Sanskrit, Chinese, Occitan and Tupi.
Nikola Tesla was capable of reading, writing, and speaking Serbo-Croatian, Slovenian, Czech, English, French, German, Hungarian, Italian, and Latin.
With our current understanding of language learning and development, were 19th-century Polyglots really proficient at these languages, or are they moreso parlor tricks for entertaining guests and diplomats? Languages are difficult to learn today, even with lifetimes of learning material available on the internet and instant audio communication available with native speakers.
We should distinguish between speaking a language and being able to read and write it. (Then I'll get to the history, I promise.)
Being able to read a language at sight is not particularly difficult. It takes intelligence and a lot of time, yes, but it isn't a rare achievement. In my field, the ability to read and (to a lesser degree) write in 5-6 languages is utterly unremarkable (it's basically a prerequisite). Being able to read 10-15 languages is far from unheard of.
Speaking that many languages, beyond the basic "Hello, how are you, I would like two oranges" is entirely different. It takes an enormous amount of time and effort to achieve that kind of spoken fluency in a language, and is basically only possible via total immersion.
With that context, the achievements of these polyglots look more reasonable. Mostly, they are talking about reading and writing languages, not speaking them.
I'll talk most about Jefferson, because I'm most familiar with him, and the languages he knew mostly overlap with those people in my field are expected to know. Jefferson was not, on the whole, speaking all of those languages. He spoke French fluently -- after he lived in France for several years. Before that point, he could read French fluently (his library contains an enormous number of French books, and he annotated them heavily) -- but when he first arrived in France, he could neither understand anything anyone else said nor be understood himself.
The same is true with the other languages Jefferson knew. We know he read Latin, Greek, Italian, etc fluently, because we have copies of his books written in those languages, and he clearly read them. A lot. Sometimes he annotated them heavily; sometimes he wrote his friends and family (who were often also polyglots) about what he was reading; sometimes he sprinkled untranslated quotations from those languages in his letters.
In fact, one of the first thing that someone who picks up older books printed in those periods might notice is just how much untranslated text and quotations they include from other languages. Take Edward Gibbon's Decline and Fall of the Roman Empire. If you get a version that includes all the original footnotes, unabridged, you'll see that Gibbon included lots of untranslated quotations in his footnotes, in languages including Latin, Greek, French, Italian, and Spanish. Not only did he read those languages (though his Greek was sometimes shaky), but he expected some at least of his readers to be able to understand those quotations as well.
You didn't mention Lord Macaulay, but I will. Macaulay was one of the biggest readers and polyglots of the 19th century. He took obsessive notes over what books he was reading when, and on what day he started and what day he finished them. His reading lists are, to say the least, intimidating. Not least because the sheer variety of languages he was reading in: on a sea voyage from Britain to India, he includes Latin, Greek, German (which he learned en route), French, and Italian.
To show that he was actually reading all those languages, and not just making up his reading lists, if you look in a copy of Macaulay's Life and Letters (edited by his nephew, and really a fun read), you'll see, again, an enormous amount of casual snippets of text in many different languages, which he expected his correspondents to understand.
Still on the subject of Macaulay, we can say something about his methods for learning all these languages, because he tells us himself. First, he would get a grammar book for the language, then read through it as he would any other book, not bothering to memorize anything nor taking notes, but just reading attentively. After that, he got a New Testament in his target language (which, he said, he knew well enough to know what it said anyway), and read through it without a dictionary. After that, he could pick up more literary works in his target language, and, now using a dictionary, work through them with some speed.
Obviously that methodology wouldn't give him any spoken fluency with the language. I suspect it also wouldn't have worked as well on a non-European language. But for learning any language that basically works the same way as languages you're already familiar with, it's a very practical way of building up a reading ability quite quickly.