There are 3 issues that come to mind:

• Trademarks - It is possible, though not always so easy, to trademark a part number. A famous example of this is where Intel produced 8086, 80186, 80286, 80386, 80486 and then Pentium. Why the change? Because you can easily trademark a largely made-up name such as Pentium. It is much harder to trademark (if at all possible) a number. If a manufacturer chooses to trademark a particular part number then, provided they enforce the trademark, only companies that license the design (or at least the name) can use the same part number.
• Patents - Patents are for a limited time - 20 years is the usual US time. While a patent is active, it may be problematic for an alternate, unlicensed, manufacturer to market a similar product with the same part number as the patented product. Either it works identically and infringes the patent or it works "similar" and confuses people and causes problems. But after all relevant patents have expired, it makes sense for an alternate manufacturer to use the original manufacturer's part number if they are producing a compatible product.
• Copyright - This gets complicated. A small IC has little to no copyright issues as long as the new manufacturer doesn't literally copy an image mask. But more complex circuits, particularly modern microprocessors, have microcode in them which can be copyrighted. Producing unlicensed versions of those chips legally typically requires clean-room reverse engineering - i.e., figure out the functionality and then write microcode from scratch to do the same thing. Unlike patents, copyrights are, at least in the semiconductor era, essentially forever.

TL;DR Once patents expire, many different manufacturers

There are 3 issues that come to mind:

• Trademarks - It is possible, though not always so easy, to trademark a part number. A famous example of this is where Intel produced 8086, 80186, 80286, 80386, 80486 and then Pentium. Why the change? Because you can easily trademark a largely made-up name such as Pentium. It is much harder to trademark (if at all possible) a number. If a manufacturer chooses to trademark a particular part number then, provided they enforce the trademark, only companies that license the design (or at least the name) can use the same part number.
• Patents - Patents are for a limited time - 20 years is the usual US time. While a patent is active, it may be problematic for an alternate, unlicensed, manufacturer to market a similar product with the same part number as the patented product. Either it works identically and infringes the patent or it works "similar" and confuses people and causes problems. But after all relevant patents have expired, it makes sense for an alternate manufacturer to use the original manufacturer's part number if they are producing a compatible product.
• Copyright - This gets complicated. A small IC has little to no copyright issues as long as the new manufacturer doesn't literally copy an image mask. But more complex circuits, particularly modern microprocessors, have microcode in them which can be copyrighted. Producing unlicensed versions of those chips legally typically requires clean-room reverse engineering - i.e., figure out the functionality and then write microcode from scratch to do the same thing. Unlike patents, copyrights are, at least in the semiconductor era, essentially forever.

This is similar to "generic" products in many other industries, with the notable exception of pharmaceuticals (due to significant safety-related regulations).

For example, Oreo is a trademark and true Oreo cookies are made only by the trademark holder and official licensees. But anyone can make a chocolate cookie with a creamy filling and sell it in a supermarket right next to actual Oreo cookies. For many uses there will be no functional difference, maybe not even in a blind taste test. But for some uses - getting a cranky child exactly the cookie they have seen advertised or eaten at their friends' homes, etc. - nothing else will do.