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Games tend to have one of two philosophies on running out of pieces: the Risk philosophy, where pieces are theoretically infinite and you can use any suitable substitute; and the Monopoly philosophy, where components are limited and you cannot take an action requiring a component if the game is out of that component.

In Monopoly itself, the rules say you cannot build more houses and/or hotels than the pieces supplied with the game. Unlimited houses and hotels is a common enough house rule that it is included as an option in many computer game adaptation of Monopoly. Implementing this house rule is fairly easy to do if you have multiple monopoly sets, and even if you don't, the even build rule means that you can indicate the build state of an evenly developed monopoly with 2+ houses by simply putting the houses/hotel on one property in the group.

So why does Monopoly have this rule in the first place? Is this just a historic artifact where early 1900's board game designers thought people would get confused about using substitute pieces or alternative representation of the game state? Is there some compelling strategic depth that this component limit adds to the game?

Keeping 4 houses on your property(ies) is a very simple strategy that reduces the length of the game quite dramatically. Tying up 8 or 12 houses prevents other players from buying those houses, keeping rents on their properties low, while increasing rents on your own. On many properties, the difference in rent between 4 houses and a hotel is relatively small and is quite easily offset by the lower rents you'll pay when landing on your opponents' properties. The moment you buy a hotel, though, those 4 (or 8, or 12) houses immediately become available for somebody else to buy.

In addition, if your opponent does buy hotels and is later forced to sell them to pay a debt, using all houses on your own property could force them to tear down all improvements. See this question for further explanation. This costs them more money to build back up and lowers the amounts you pay in the meantime.

Using the "unlimited houses/hotels" house rule is probably the number one mistake people make when trying to play a short game. Computer games against AI can be played in 15-30 minutes using the correct rules; 4 player human games will take a bit longer due to thinking time and socializing, but should stay well within an hour. Hours-long games are almost entirely due to house rules preventing people from losing money as quickly as they otherwise would.

From a thematic standpoint, the precursor to Monopoly was called "The Landlord's Game" and it was designed by the socialist Elizabeth Magie in order to demonstrate the evils of capitalism - the people who own property get richer and more able to buy property, while the people who have to rent just keep losing money and eventually run themselves into the ground.

While I couldn't find any concrete evidence that this was deliberate, having a limit on the number of houses helps to represent the fact that property owners who are able to develop the land they own can quickly recoup their costs, while at the same time making it harder for others to do the same (because of limit amounts of resources with which to build, etc).