Why would Patrick Bateman share a phone number with Paul Allen and Timothy Bryce?

The number on the card is almost certainly the switchboard / operator / company directory number for the company they work at. When you call that number, you are then asked where you want your call directed. (Possibly by a person, possibly by an automated number).

In some cases, business cards might also include an extension number for you to dial directly, but not always.

In this case, it's possible that the company doesn't want clients direct-dialing their executives. Perhaps the executives are out of the office a lot, or perhaps the company wants to maintain some control over who gets access the their officer-level employees (in case one leaves, or is on vacation, or whatever).

So the business cards only have enough information on them for a client to get in touch with the company's operator and tell them who they want to speak to, and the operator handles it from there.

While everybody else's answer may be correct in the context of reality, you're missing the point of the scene, if not the entire movie.

He works at the same company as everybody else, has the same position, shares the same lifestyle, the same interests and hobbies, has the same clothes, the same routines and hangouts, they share the same women, have similarly expensive apartments...

His entire "mask of sanity" is based on possessing all of the same things his peers have.

Why wouldn't he have the same number as everybody else? Bateman doesn't actually have an identity of his own. His existence revolves around emulating his peers to fanatical ends. The phone number is just another thing they all have in common, a serial number that fails to differentiate any of these characters from one another.

Some other answers make the point that Bateman's company would likely have operated a switchboard; in which case someone calling the company would connect with an operator who would forward them to the appropriate party. While this may be the case for actual institutions, I don't think it applies to this scene. Rather, I think this was a complete technical oversight in the movie (unless Johnny's answer is correct which is a pretty interesting possibility).

Consider the conversation between Bryce and Paul Allen:

Bryce: Listen, Paul. Squash.

Paul Allen passes Bryce his card business card.

Paul Allen: Call me.

It is clear from this exchange that Paul Allen gives Bryce his card so that Bryce could have his phone number to call him later about playing squash. There is no reason that Paul Allen would have needed to give Bryce his card if he and Bryce had the same phone number, but as we see on both of their cards, they do. Assuming this wasn't done intentionally for symbolic reasons, this can be nothing other than an oversight.

In the UK, banks and large financial institutions need to comply with the Financial Conduct Authority's rules regarding counterparty call recording, basically making sure that all calls relating to deals and conversations with the company's legal representatives are recorded for compliance and audit purposes. This is usually accomplished by having a single line in and out of the company through which 'official' calls are routed to clients.

Many large companies (UK and US) have a single switchboard so that they can monitor who is making and receiving calls to their clients. A department with responsility for multi-billion dollar mergers would certainly want this ability.