It's farcical to imagine a tony professor and a warehouse worker hold the same relationship to capital and society. If the entire powerbase of the capitalist class in human terms were just the 200,000 car dealership owners, 15,000 bank owners, Jeff Bezos, and Elon Musk, they would be eaten overnight. There must be a broader base of support built through material conceptions and reinforced through superstructural elements or "social signifiers".

These two groups do not "have the same relationship to the means of production." one section is exploited in the main, the other is mixed and facilitates greater exploitation of others in exchange for security, comfort, and social prestige.

What do you mean by the Marxist conception? Marx himself sometimes uses the term middle class.

Here's a few examples.

The Communist Manifesto, Chapter 1:

The lower middle class, the small manufacturer, the shopkeeper, the artisan, the peasant, all these fight against the bourgeoisie, to save from extinction their existence as fractions of the middle class. They are therefore not revolutionary, but conservative. Nay more, they are reactionary, for they try to roll back the wheel of history. If by chance, they are revolutionary, they are only so in view of their impending transfer into the proletariat; they thus defend not their present, but their future interests, they desert their own standpoint to place themselves at that of the proletariat.

The 18th Brumaire of Louis Bonaparte, Chapter 1:

The bourgeois monarchy of Louis Philippe can be followed only by a bourgeois republic; that is to say, whereas a limited section of the bourgeoisie ruled in the name of the king, the whole of the bourgeoisie will now rule in the name of the people. The demands of the Paris proletariat are utopian nonsense, to which an end must be put. To this declaration of the Constituent National Assembly the Paris proletariat replied with the June insurrection, the most colossal event in the history of European civil wars. The bourgeois republic triumphed. On its side stood the aristocracy of finance, the industrial bourgeoisie, the middle class, the petty bourgeois, the army, the lumpen proletariat organized as the Mobile Guard, the intellectual lights, the clergy, and the rural population. On the side of the Paris proletariat stood none but itself.

Capital Volume 1, Chapter 25, Section 4:

Pauperism is the hospital of the active labour-army and the dead weight of the industrial reserve army. Its production is included in that of the relative surplus population, its necessity in theirs; along with the surplus population, pauperism forms a condition of capitalist production, and of the capitalist development of wealth. It enters into the faux frais of capitalist production; but capital knows how to throw these, for the most part, from its own shoulders on to those of the working class and the lower middle class.