Yes, ish, but I think it's a bit more nuanced than that, and comes down in part to distinctions within social classes.
This is Kate Fox's Watching the English on social class and flooring (she doesn't address 'shoes on/off')
...the way in which we arrange, furnish and decorate our homes is largely determined by social class. This has little or nothing to do with wealth. Upper-class and upper-middle-class homes tend to be shabby, frayed and unkempt in a way no middle-middle or lower-middle would tolerate, and the homes of the wealthiest working-class nouveaux-riches are full of extremely expensive items that the uppers and upper-middles regard as the height of vulgarity. The brand-new leather sofas and reproduction-antique dining chairs favoured by the middle-middles may cost ten times as much as the equivalent items in the houses of upper-middles, who despise new leather and ‘repro’.
In the homes of the middle-middles and below, the ‘lounge’ (as they call it) is more likely to have a fitted carpet (among the older working classes, this may be a patterned carpet). The higher castes prefer bare floorboards, often part-covered with old Persian carpets or rugs.
UC and UMC would regard asking a guest to take off their shoes as a minor social crime -- as nouveau riche, gauche and anxious about protecting a house that should be robust, lived-in and above minor damage, even if the Persian rugs are priceless. Shabby gentility should be able to cope with dirt.
LMC and WC are more likely to ask visitors to take their shoes off as there's often a different, very houseproud idea of the home, as immaculate private sanctuary from the grubby outside world. Also, traditionally, a fitted carpet is likely to have been a significant financial outlay, which needs protecting. (See also leaving the plastic covers on sofas or three-piece suites.)
Having said that, Kate Fox also acknowledges that the influence of interior design programmes has altered flooring choices
Thanks to the influence of the upper-middle-class interior designers who dominate the many home-improvement programmes on television, some formerly clear class-indicators are no longer so reliable. Middle-middle, lower-middle and even working-class homes may now have wooden floors rather than fitted carpets (and even the naffest of nouveaux riches now avoid the deep-pile variety).
But the idea that the pristine floor must be guarded and protected has lingered, even with wooden flooring that can be mopped clean quickly and easily.