Communication between classes always represents a critical situation for the language that is used, whichever it may be. It tends to provoke a return to the sense that is most overtly charged with social connotations: 'When you use the word paysan (peasant) in the presence of someone who has just left the countryside, you never know how he is going to take it.' Hence there are no longer any innocent words.

Pierre Bourdieu, Language and Symbolic Power, 1990.

First, some disclaimers, the medieval era is a long period of time with no strict definitions (though 500-1500 roughly covers most start and endpoints), but it also generally comes with a geographic boundary; it's mostly limited to the Mediterranean region: Europe, North Africa, and the Middle East. It is basically the region that was once under direct control or heavy influence of the Roman Empire following the collapse of the Western Roman Empire (or in some definitions, the Eastern Roman Empire’s crippling by the Muslim Caliphate) and prior to the modern era, marked either by the fall of Constantinople to the Ottoman Empire or the beginning of European contact with the Americas.

My background is in applied linguistics and centered on the languages I speak: Spanish, English, and French (though I can stumble my way through Latin). So, narrowing the context down even further, my answer will mostly be limited to western Europe, mostly France and England, in this period, where the languages I speak developed into their modern forms. I'll try to address some of the fundamental ideas here, so hopefully, it will also be applicable to thinking about how language works as a manifestation of power in other contexts.

Now, that last line might have given you an idea of where my answer is headed. When we are talking about an elite being marked out by their use of language, we are talking power and capital. And, in my opinion, there is no better source for exploring this topic than the work of French sociologist Pierre Bourdieu (RIP, the 20th anniversary of his passing is next Sunday). So, before looking at some specific examples, let's see what Bourdieu has to say about language and capital.

In The Forms of Capital (1986), Bourdieu defines capital as "accumulated labor (in its materialized form or its “incorporated,” embodied form) which, when appropriated on a private, i.e., exclusive, basis by agents or groups of agents, enables them to appropriate social energy in the form of reified or living labor". Basically, it is a way of storing labor in a way that allows the user to effect some change on the world.

Bourdieu identifies three fundamental forms of capital, quote

economic capital, which is immediately and directly convertible into money and may be institutionalized in the form of property rights; as cultural capital, which is convertible, in certain conditions, into economic capital and may be institutionalized in the form of educational qualifications; and as social capital, made up of social obligations (“connections”), which is convertible, in certain conditions, into economic capital and may be institutionalized in the form of a title of nobility.

Mastery of a particular language variety falls into the category of cultural capital. In the first chapter of Bourdieu's Language and Symbolic Power, the idea of language as capital is illustrated by way of a quote from Claudel's Le Soulier de satin (The Satin Slipper).

As you say, my good knight! There ought to be laws to protect the body of acquired knowledge. Take one of our good pupils, for example: modest and diligent, from his earliest grammar classes he's kept a little notebook full of phrases.

After hanging on the lips of his teachers for twenty years, he's managed to build up an intellectual stock in trade; doesn't it belong to him as if it were a house, or money?'

In our modern world, a lot of cultural capital is more institutionalized. Usage of the standard language is a prerequisite to most educational qualifications and though "university English" isn't identical to bourgeois varieties, it is far closer to them. When was the last time you read a doctoral thesis that made use of habitual be?

However, even outside of our more institutionalized expressions of cultural capital, we have, as you pointed out, preconceptions about speakers based on how they speak; the wealthy speak a certain way. In linguistics, we call the value associated with a language variety used by elites prestige. However, there is also covert prestige; we can find, throughout history, a certain admiration for, and often appropriation of, elements from speech associated with non-elites.

The linguistic situation of the medieval world is at times more opaque, for many reasons, partly because there was less institutionalization of cultural capital, at least along the lines we would think of it nowadays. It was also different in how it defined "a language". Our modern languages are standardized and often clearly tied to the identity of a nation, this was not so before the modern era. In the medieval era, there was no need for such standardization, as Bourdieu points out:

so long as a language is only expected to ensure a minimum of mutual understanding in the (very rare) encounters between people from neighbouring villages or different regions, there is no question of making one usage the norm for another (despite the fact that the differences perceived may well serve as pretexts for declaring one superior to the other)

Before the French Revolution, for example, France did not have a language; instead, various local varieties which we can group under umbrellas such as languages or dialects were used, though the speakers of these varieties wouldn't have necessarily thought of such groupings as relevant or accurate. The same was roughly true for the rest of Europe during the period in question and it's still true to a certain extent in regions of the world where the modern western concept of the nation-state has not fully taken hold.

I give you all of this background knowledge so that I can directly answer your question with a resounding YES. Yes, a medieval royal's speech variety would have served to mark them as different from a common peasant. People are always driven to reaffirm their identity through language and elites in the medieval era would have been no different. They would have sought to establish their form of speech as one that carried with it a sense of authority.

But they likely wouldn't have thought of this along the lines of accent or dialect. It also wouldn't have been necessarily institutionalized in the same way standard language is today. That is, though you would have heard different types of speaking from a royal and a peasant, in general, there would have been no expectation that the peasant educate themself to speak more like the royal or like a perceived standard. They were a peasant, who spoke as a peasant did.

This also meant that there was no real expectation that they even speak the same "language", which they very often didn't. The best example here is Norman England. When the Duke of Normandy, William the Bastard Conqueror took control of England in 1066, he carried out an extremely thorough purge of the English-speaking nobility and replaced them with Normans (in the main), who spoke a variety of Old French known to us as Norman French. The variety of French that developed in this community is known as Anglo-Norman.

For centuries following the conquest, Anglo-Norman was the language of the elites, and comprehension of English was spotty at best. Many Anglo-Norman terms filtered down into English, some examples of the nature of this exchange can be seen in food terms. Though the English names for animals such as cow, deer, and pig persisted, the food products from these animals come to us from Anglo-Norman, beef, venison, and pork.

Though the social image of the era is not as well-recorded as we might like, you can imagine the complex web of languages and dialects a speaker might have to navigate. A good comparison for most of medieval England (especially outside of major urban areas) might be modern-day Puerto Rico, with English acting as the language of a foreign elite who rule over the island, but the internal varieties of Spanish remaining strong markers of both class and region within the Spanish speaking community.