UPDATE - found actual in-universe canon confirmation
The Most Radiant Gesar, aka Gesser, aka Djoru the Snotty, aka Boris Ignatievich, ... the Most Radiant Gesar, Higher Magician and Magician Beyond Classification, Light Other, Conqueror of Demons and Son of Heaven, Hero of Tibet and Mongolia, central character of the national epic the Gesariada, venerated by the Kalmyks and honored with a huge equestrian statue in Buryatia, head of the Night Watch of Moscow and therefore, de facto, of the whole of Russia, was standing behind me.
(Lukyanenko, "Sixth Watch", Andrew Bromfeld translation. Part 1, "Mandatory Actions", Chapter 3.)
Real life legend summary
According to both Wikipedia entry on Gesar, as well as serious academic research of Geseriad, Geser's childhood (as per legend) is described thusly:
Born in Tibet
Snotty as a child.
Had an enemy named Soton/Tzoton.
Was a Trickster archetype
Super-powerful mage (basically, a reincarnated deity) fighting the forces of evil
I will provide details of these 5 real-life points and citations below, but first, let's see how well in-universe Gesar matches:
In Book #5 of Watches cycle, "The New Watch", we see the hero read a children's textbook, which basically paints a "true in-universe biography of Gesar" as matching these very four details from our real-life folk legend:
... Издательство "Иное слово". Печатает книги и прочую бумажную продукцию для Иных. Для Светлых и Темных, без разбора. ... Книжка называлась "Детство замечательных Иных". ... Для внеклассного чтения. Истории про детство великих волшебников. ...
... Наткнулся на статью про Гесера и ухмыльнулся, прочитав первые строчки:
"Когда великий Гесер был маленьким, он жил в горах Тибета.
... его даже обзывали обидным именем Джору – "сопливый". ...
Про это знал только Темный Иной – Сотон, который мечтал сделать Гесера Темным…"
("Новый Дозор", Часть первая "СМУТНЫЕ ЦЕЛИ", Глава седьмая)
... Published by "Other word". They print for Others. For Light and Dark, indiscriminately. ... The book was called "Childhood of remarkable Others". ... For outside-classroom reading. Stories about the childhood of the great wizards. ...
... I stumbled on an article about the Gesar and smirked when reading the first lines:
"When Great Gesar was young, he lived in the mountains of Tibet.
... He was even insultingly called "Joru" - meaning "snotty". ...
Only the Dark Other - Soton - knew he was Other, and wanted to make Gesar Dark ... "
("The New Watch", Part One "Vague Goals", Chapter Seven)
As we can see, all 4 first points are matching very precisely.
Of course, the fact that in-universe Gesar was a trickster was fully explored in SFF Q&A " Is Gesar a joker? ".
Um, do I really need to prove that he's magical in-universe and fights the forces of evil? Didn't think so.
So, Gesar in the books matches the real life Geser on all 6 points.
My original answer version merely cited Wikipedia, but for a more epic (pun intended) answer, I decided instead to look at...
Actual academic research:
One of the best academic resources I could find on Geser that had a full e-version was a Russian book "The History of the World literature, Part 3", published in 1985 by Georgy Petrovich Berdnikov, on pages 362-364, chapter "Tibetian-Mongolian Geseriad" (Автор: Бердников Георгий Петрович - Книга: "История всемирной литературы Т.3" - Страница 362-364, "ТИБЕТО-МОНГОЛЬСКАЯ ГЭСЭРИАДА")
Berdnikov cites specific published legend version of the legend (1716 Beijing version, the first printed one), but also analyzes other versions.
Героическое детство Гэсэра наиболее полно разработано в ксилографической (пекинской) версии 1716 г...
Heroic childhood of Geser is developed most fully in xylographic (Beijing) 1716 version...
One other reason (aside from e-book availability) that I rely on this book more heavily (instead of more specialized academic research like R.A.Stein or Alexandra David-Neel), is that the original SFF story by Lukyanenko is Russian, which means any real-life epic references he used were likely in Russian and not English or French, which in turn means he's far more likely to have been using Berdnikov.
All citations below are from this book, unless specified otherwise.
Born in Tibet (specifically, Ling kingdom)
“Somba Yexei Banjor is a scholar who opinion is that Gesar was a real person who lived in the Ling area of Kham (source).
Seriously, any serious reference mentions Tibetian origin so it's useless to cite any more, but why not cite Berdnikov again? :)
С XI в. в тибетских текстах встречается упоминание о Кесаре, но не Кесаре из Лина (упоминания о последнем дошли лишь от XVII в.), а Кесаре из Крума (Фрума) (Бердников)
From 11th century, Tibetian texts mention Cesar, but one from Krum and not Linn. (Berdnikov)
Сам Гэсэр до поры до времени выступает в виде никчемного и вместе с тем проказливого Цзуру (тиб. Чжору). ... (Бердников)
Geser himself for the time being acted as worthless, yet mischievous Tzuru (Tibetian Choru). ... (Berdnikov)
Or, here's the summary of the epic being performed:
How young Choru was expelled from Ling Kingdom by his jealous uncle Trothung, but thanks to his magical skills and a famous horse race became the King of the Ling and married queen Dugmo
Also, see the citation from Stein in #3.
Snotty as a child.
... Важнейшие элементы сказания о Гэсэре, ... детские подвиги и проделки — часто под видом никчемного "соплячка", — преследование злого дяди (Тхотуна, монг. Цотона)... (Бердников)
... The most important elements of Geser epic ... childhood feats and pranks - often under the guise of worthless "snotty kid" - the pursuit of his evil uncle (Thotun, Mongolian Tsoton) ... (Berdnikov)
Another citation (which also has independent confirmation of Choru/Joru name)
... The hero's life is divided into two parts... the hero even bears different name in each of them. In his youth he is Joru, an ugly, snotty-nosed lad...
"Tibetan Civilization" By Rolf Alfred Stein, 1962/1972, p.280
The same is likely noted in same author's French book "Recherches sur l'épopée et le barde au Tibet, Volume 13" from 1959, but I couldn't find the text to cite; but Wikipedia's cites the snottiness from this book, page 545.
Had an enemy named Soton.
See the quotes from #3 above, as well as:
В годы детства Гэсэра родители его находятся в изгнании и бедности по вине Цотона.
Целый ряд проделок он совершает для наказания алчного и злобного Цотона.
During Geser's childhood, his parents are in exile and poverty, due to Tsoton.
A number of tricks he performs are to punish the greedy and evil Tsoton.
Tzoton is mentioned in various other places in Berdnikov; and in Stein; as well as a quote I used in #2 above from the description of the video of the epic's performance, not worth citing them all.
Was a Trickster archetype
... архаических чертах мифологического озорника (ср. ... Локи в «Эдде»), переплетенных с чертами шамана-волшебника и настоящего воина-богатыря.
... archetypical features of mythological Trickster (similar to... Loki in the "Edda"), interwoven with features of shaman-wizard and a true warrior-hero.
Also see quote from #3 above.
A lot more detail can be found in a book that contains a very very detailed analysis and summary of the Geseriad, by a researcher of the topic, and which is frequently cited by others: "The Superhuman Life of Gesar of Ling" By Alexandra David-Neel, translated from French.
It's hard to offer citations as the book isn't freely visible on Google Books, but it definitely confirms point #4 (Tsoton as enemy) and #5 (trickster) even from small snippets available.
Full list of references
Lukyanenko, Sergey, "The New Watch" - book #5 of The Watches cycle (Сергей Лукьяненко, "Новый Дозор")
Is Gesar a joker? @ SFF.SE
Russian Wikipedia entry on Geser (Гэсэр)
English Wikipedia entry on Epic of King Geser
"The History of the World literature, Part 3", published in 1985 by Georgy Petrovich Berdnikov, pages 362-364, chapter "Tibetian-Mongolian Geseriad" (Автор: Бердников Георгий Петрович - Книга: "История всемирной литературы Т.3" - Страница 362-364, "ТИБЕТО-МОНГОЛЬСКАЯ ГЭСЭРИАДА")
"The Superhuman Life of Gesar of Ling" By Alexandra David-Neel, Yongden, Lama - first published in 1933, translated from French
Stein, Rolf A. (1959). Recherches sur l'épopée et le Barde au Tibet. Paris: Presses Universitaires de France.
Buryat version of the epic translated into English - unfortunately, that version doesn't go into Geser's childhood the way Mongol version does.