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.