It is a multifaceted question and without an easy answer. The Grand Duchy of Lithuania was a multinational state which encompassed much of the former Kievan Rus and left an unquestionably massive legacy onto the region, so it is understandable that its legacy can be interpreted in different ways. In Russia, there was a sort of revival of the memory of the Grand Duchy of Lithuania in the 19th century, during which the idea of "Lithuanian-Russian" Lithuania, or "Lithuanian Rus" (Литовская Русь) were first formulated, and were firmly established in Russian academia by 1896, when the "Lithuanian-Russian State" was included into the Brockhaus and Efron Encyclopedic Dictionary (see here). The context of the 19th century helped this interpretation of Lithuanian history to form. Prior to 1795, there was a clear and sharp distinction drawn between "Lithuanian" and "Moscovite" Rus', as these two nations were long standing enemies and regional rivals, but in 1795, Lithuania was abolished alongside the rest of the Polish-Lithuanian republic and its territory was annexed by Russia. The concept that prior to the Polish-Lithuanian union, Lithuania was a nation jointly created by Lithuanians and East Slavs and that Lithuania was thus an East Slavic state which was merely separated from its brethren in Moscow by Polish influence was thus a way to legitimize the annexation of modern day Lithuania, Belarus and Ukraine.

On the other hand, Lithuania is, to this day, an important source of national identity for the modern Belarusian and Ukrainian nations, and the means via which both of them seek to draw their distinction from Russia. The campaign by Grand Duke Algirdas against the Golden Horde in 1361-1363, culminating in the Battle of Blue Waters, is seen as the liberation of Kiev from the Mongol yoke, the Grand Dukes of Lithuania are seen as equally Belarusian/Ukrainian as they were Lithuanian, receive memorials and monuments across both nations (for example, Lutsk, the last residence of Vytautas and the place where he was expected to be coronated in 1430, has a memorial of the Vytis on the walls of the Lutsk Castle and the restaurant "Korona Vitovta", Vitebsk has recently erected a statue for Grand Duke Algirdas, etc.).

How founded is it in reality? It cannot be denied that the Grand Duchy of Lithuania was very, very closely connected to Ruthenian culture and for hundreds of years, the history of Belarus and Ukraine was the history of Lithuania, and there are plenty of arguments to make that Lithuania was equally built by the Slavs as it was by the Lithuanians:

• The chancellery language of Lithuania, in which some of its most important documents, such as the Statutes of Lithuania and the Lithuanian Metrica, were written, was Chancery Slavonic, or simply Ruthenian.
• The first and only King of Lithuania, Mindaugas, was allegedly crowned in Novohrudak and the city was his first capital, sometimes it is also claimed that he was originally baptised as an Orthodox (though this is heavily disputed and sources for these claims are largely nonexistent)
• Numerous princes and relatives of the House of Gediminas, the ruling dynasty of Lithuania during its height, settled down in Ruthenian territory, converted to Orthodoxy, and were the progenitors of later powerful Ruthenian noble families.
• Lithuanian invasions into the former Kievan Rus' generally retained the existing legal and social systems, there was no significant change of nobility or governance, princes and dukes who submitted to the Grand Dukes of Lithuania peacefully were allowed to retain their lands and power.
• In the 15th century, when the official titles of Lithuania were being slowly codified, "Grand Duchy of Lithuania and Rus'/Ruthenia" was among the most prominent and became effectively official with the Statutes of Lithuania, in which the state was defined as the "Grand Duchy of Lithuania, Rus' and Samogitia". This title preference has precedent in the preceding centuries, such as in the letters of Grand Duke Gediminas in 1323, in which his seal is described to have written "Gediminas (Gedeminni), King of Lithuanians and Ruthenians".

In Lithuania, which I am personally more familiar with, medieval Lithuania is seen as primarily a Lithuanian state. The situation in Lithuanian historiography has gone to the better since independence, as past interpretation of it being solely Lithuanian, with Ruthenians and other peoples as merely conquered peoples, has been mostly abandoned in favor of a pluralistic approach where the contribution of East Slavic culture, political thought, legal systems, institutions and religion into medieval Lithuania is recognized and it is accepted that the Ruthenians saw Lithuania as theirs, though it is still emphasized that Lithuania was a Lithuanian state:

• Lithuania was founded in the 13th century in Lithuanian-speaking territory which belongs to Lithuania in the modern day, its capitals during its history - the hypothesized "Voruta", Kernavė, Trakai and Vilnius - are all placed in a region which, up to the 13th century, was unquestionably a pagan, Baltic language speaking territory.
• In spite of the lax approach towards local institutions, traditions and religion, there is a very clear distinction drawn between "Lithuanians" and "Ruthenians" in contemporary writings. The history of medieval Lithuania to 1385 is indistinguishable from its status as the only major pagan state in Eastern Europe which had to wage a war of destruction against the Order-State up until its christianization and thus subsequent union with Poland. Lithuania is explicitly defined as a pagan state in both Teutonic and East Slavic contemporary writings, which means that its rulership and court, at minimum, retained their pagan faith, and were thus likely Lithuanian. The Ruthenian population of the former Kievan Rus' on the other hand, was almost universally Eastern Orthodox Christian.
• Finally, the correspondence of Lithuanian dukes with the Teutonic Order shows that they made a connection between their background and the background of the Baltic-speaking peoples of Prussia and Samogitia. Grand Duke Algirdas, discussing terms of his possible Christianization during his reign, consistently emphasized that the territory of Prussia was theirs by right and should be returned to them if peace were truly established between the Order-State and a hypothetical Christian Lithuania. More famously, Grand Duke Vytautas argued in a 1420 letter to Sigismund, Holy Roman Emperor, regarding the fate of Samogitia, a region disputed between Lithuania and the Order-State, by writing that "[...] first of all, you made and announced a decision about the land of Samogitia, which is our inheritance and our homeland from the legal succession of the ancestors and elders. We still own it, it is and has always been the same Lithuanian land, because there is one language and the same inhabitants.". Further onwards, he describes that the names of the two regions of Lithuania, Žemaitija (Samogitia) and Aukštaitija, originate from "low" (žemai) and "high" (aukštai), meaning "lowland" and "highland" of one "Lithuania", respectively.