How reliable is it to use another manufacturer's datasheet of the same part?
Generally, the primary specifications will be the same, but beware as the devil is in the details as noted by Olin.
As an example, take the LM1117.
Parts with the same base number are also made by On Semiconductor and AMS.
Looking at the datasheets, TI has this to say on stability:
The output capacitor is critical in maintaining regulator stability, and must meet the required conditions for both minimum amount of capacitance and equivalent series resistance (ESR). The minimum output capacitance required by the LM1117 is 10 μF, if a tantalum capacitor is used. Any increase of the output capacitance will merely improve the loop stability and transient response. The ESR of the output capacitor should range between 0.3 Ω to 22 Ω. In the case of the adjustable regulator, when the CADJ is used, a larger output capacitance (22-μF tantalum) is required.
AMS simply states:
Stability
The circuit design used in the AMS1117 series requires the use of an output capacitor as part of the device frequency compensation. The addition of 22μF solid tantalum on the output will ensure stability for all operating conditions. When the adjustment terminal is bypassed with a capacitor to improve the ripple rejection, the requirement for an output capacitor increases. The value of 22μF tantalum covers all cases of bypassing the adjustment terminal. Without bypassing the adjustment terminal smaller capacitors can be used with equally good results. To further improve stability and transient response of these devices larger values of output capacitor can be used.
On Semiconductor has this:
Frequency compensation for the regulator is provided by capacitor Cout and its use is mandatory to ensure output stability. A minimum capacitance value of 4.7 μF with an equivalent series resistance (ESR) that is within the limits of 33 mΩ (typ) to 2.2Ω is required. See Figures 12 and 13. The capacitor type can be ceramic, tantalum, or aluminum electrolytic as long as it meets the minimum capacitance value and ESR limits over the circuit’s entire operating temperature range. Higher values of output capacitance can be used to enhance loop stability and transient response with the additional benefit of reducing output noise.
You should note that all these statements have subtle differences for a part that is designed for the same task; other parameters in the datasheets vary as well.
This is but one type of part from the millions out there. Even the humble resistor and capacitor from various manufacturers can have differences (even though they are apparently the same type of device) that you may care about (in high reliability designs, this is definitely true).
Update.
Dim makes an excellent point on schematic notation where the generic number may not be sufficient.
In what I currently do (avionics including flight controls) we have internal part numbers which are used in the schematic; these numbers map to a single part from a single manufacturer to deal with this precise issue.
If you are using a specific manufacturers part, use that manufacturers datasheet.
No, using one manufacturer's datasheet to get specs on another manufacturer's part is not a good idea, even for common parts like a 1N4148.
In this case, there should be the minimum specs somewhere for what a 1N4148 must do, but individual manufacturers may exceed those. You can only count on the minimum specified in the generic 1N4148 definition when using parts across manufacturers. Unfortunately, finding that minimum generic spec is often not easy.
For a 1N4148 in particular, I've seen quite a range of maximum continuous forward current across manufacturers. The things shown in graphs are usually not specs anyway, so relying on them is not a good idea in general. And, these are the characteristics most likely to differ between manufacturers.
Don't do it.
The manufacturers are going to try and match the specs, but no they won't be exactly the same. Think about it, Manufacturer A comes out with a part, and Manufacturer B reverse engineers it or matches the specs with their own design. So, no you aren't going to get the exact same physical results between devices, most of the time they are physically close to being the same but are not the same.
One thing to note: A more detailed datasheet means that they did more testing and possibly more factory testing, usually specs in the datasheet are guaranteed to be consistent from the devices manufactured and some manufactures actually test all devices to make sure they are consistent with advertised specs. Some companies rip off designs from other manufacturers and don't do the testing (but are cheaper parts), and you can see this reflected in the datasheets. The adage "you get what you pay for" can be applied to electronic parts.