How critical is it to declare the manufacturer part number for a component within the BOM?
It obviously depends from the component and what it is used for in a circuit. Take for example a simple 1k resistor in a certain package.
A manufacturing facility would have component engineers that deal with alternative parts, and since they cannot know what the resistor does in a circuit, they also have no clue what kind of resistor they should select. So if you don't specify at least one specific resistor part or multiple different specific parts, they don't take the risk and put just something there, they will ask you for instructions suggest alternatives that you must choose from.
Sometimes, it could be that any 1k resistor with 10% tolerance for a LED is fine, but if it needs to be a precision 1% resistor for some sensitive thing, it makes a difference. Also for high speed AC circuits the resistor material makes a difference, a wirewound resistor won't work while a film resistor would. For low noise a carbon resistor may be out of the question. For high pulse current applications a standard resistor is not acceptable. There are just so many different kinds of 1k resistors, even if you did state the tolerance, package, wattage, and many other things.
And no, the part numbering does not mean that chips with identical part from different manufacturers behave identically under all circumstances. There might be a reason why you chose a specific manufacturer. Or if there is not, then indicate it that it should not matter. But then you should also be sure that any manufacturer is fine, perhaps by looking at datasheets and/or testing the component in the circuit before accepting the use in design.
Sometimes even a same chip but from different batch or with different silicon revision fails to work as intended in a circuit.