Well, yes and no. Timur saw himself in many ways as continuing the Chinggisid legacy, but pointedly refused to take the title of khan, which by this time was only borne by descendants of Genghis Khan (and indeed, ruled through puppet khans of Chinggisid descent for much of his career). The "official" titles which he most often used to describe himself were amir, or ruler, and küregen (gūrkān), or "son-in-law," reflecting his marriage to a number of Chinggisid princesses. (Which is why the Mughals, themselves descendants of both Timur patrilineally and the Chinggisids matrilineally, usually called themselves by the name of Gūrkānī.)

That being said, Timurid genealogical traditions portrayed the Barlas—Timur's own clan group—as a parallel to the Borjigin in many respects, and created a role for Timur's ancestor Qarachar Barlas as an advisor to Genghis Khan and his son Chaghatai. While the most complete examples of this tradition date after Timur's death, it has been convincingly argued that these have their origins in work done during the conqueror's own life. Most historians these days would accept that these genealogical practices were initiated by Timur as part of his program of defining the Barlas clan as historical assistants to the Chinggisid monarchs (as indeed he himself pretended to be).

Timur certainly drew some criticism for allegedly ruling more as a Chinggisid than a Muslim. The Damascene historian Aḥmad b. ʿArabshāh wrote of him:

He clung to the laws of Chinggis Khan, which are like branches of law from the faith of Islam, and he observed them in preference to the law of Islam. Thus it is also with all the Chaghatays, the people of Dasht, Cathay, and Turkistan, all which infidels observe the laws of Chinggis Khan, on whom be the curse of God! rather then the laws of Islam and accordingly our mawlana and shaykh, Hafiz al-Din Muhammad al-Bazzazi, on whom be the favor of God! and our mawlana, Sayyidna and doctor, ‘Ala’ al-Din Muhammad al‑Bukhari, whom God preserve! and other doctors and banners and leaders of Islam have given a response to all, that Timur must be accounted an infidel (kafir) and those also who prefer the laws of Chinggis Khan to the faith of Islam, and also for other reasons.

And goes on to favorably compare Timur's son Shah-Rukh to him as a "legitimate" Muslim ruler. While this claim as a whole is open to debate—Beatrice Manz, for example, argues that Shah-Rukh's reign saw the fullest co-opting of Chinggisid legitimation by the Timurid state—it is certainly true that on certain issues, the former tended to maintain some "traditional" Chinggisid policies which theoretically ran counter to religiously acceptable practices. Taxation, for example, was a major issue, and the tamgha, a customs tax levied on goods entering or exiting a given domain, was vehemently criticized by those who wanted to see a more Islamic style of rule. Timur himself had been a firm proponent on the tamgha, while Shah-Rukh saw it canceled, and thereafter positions oscillated with rulers, and sometimes changed within the space of a single reign.

So as far as political legitimacy goes, Timur and his descendants were certainly aware of and used Chinggisid strategies (though again, Timur himself drew the line at claiming full sovereignty). Genealogically, though, they did not consider themselves Chinggisids. Some genealogical traditions do hint at a common ancestry for the Barlas and the Borjigin, but enough generations back that Timur certainly held no illusions of descent from Genghis Khan. (Other traditions played up other genealogical claims, e.g., descent from ʿAli.) As for "the people living in Timur's empire," well, some of them were Mongols, and they certainly considered themselves descendants of the same! Others not ethnically Mongol could certainly trace their families back to tribal groups who had accompanied the Mongols. But there's no evidence of a widespread belief that most people in Timur's domains were Mongol in origin, if that's what you mean...?

Two articles that you may find relatively helpful here:

Manz, Beatrice Forbes. "Temür and the Problem of a Conqueror's Legacy." Journal of the Royal Asiatic Society 8, no. 1 (1998): 21-41. [JSTOR—note that JSTOR is giving free read-online access through the end of December, so even if you don't have institutional access you could still read this!]

Woods, John E. "Timur’s genealogy." Intellectual Studies on Islam: essays written in honor of Martin B. Dickson (1990): 85-125.

(General bibliographic note: Manz, Woods, and Maria Subtelny all have done a decent amount of work on the Chinggisid-Timurid tradition and are worth looking into.)

Compare Woods with Morimoto, Kazuo. "An Enigmatic Genealogical Chart of the Timurids: A Testimony to the Dynasty’s Claim to Yasavi-ʿAlid Legitimacy?." Oriens 44, no. 1-2 (2016): 145-178.

Finally, I haven't read it yet, but a dissertation from Toronto written this year focuses on many of the themes you were asking about and may be worth a read:

Kuang, Shuntu. "For King and What Country? Chinggisid-Timurid Conceptions of Rulership and Political Community in Relation to Territory, 1370–1530." PhD diss., 2020. [link]