Is it possible to have two different makes of engine on a single twin-engine aircraft?
Yes it is possible.

For example :


  A Boeing 777-200ER using GE engines . The first engine = GE90-94B and the second engine = GE90-115B .


This is called engine intermixing.


  On 737 sometimes The more powerful engine is derated to the same level as the other engine, and we had to do full power takeoffs everytime. As long as the FMC is aware of the intermix, pilots are allowed to use autothrottle.


Here are the sources for your further reading : 
Source 1
Source 2

Following is another example:


  Models with engine intermix already certified


Aircraft models / Engines        Engines allowed

A310-221 / PW JT9D-7R4D1         1 or 2 JT9D-7R4E1
(48000 lb *)                     50000 lb*
                                 (A310-222)

A310-222 / PW JT9D-7R4E1         1 or 2 JT9D-7R4D1
(50000 lb *)                     48000 lb *
                                 (A310-221)

A310-308 / GE CF6-80C2A8         1 or 2 CF6-80C2A2
(57860 lb*)                      52460 lb*  
                                 (A310-304)

A300B4-605R / GE CF6-80C2A5      1 or 2 CF6-80C2A3
(60200 lb*)                      58950 lb*
                                 (A300B4-603)

Note (*): Static thrust limit at sea level for takeoff with OAT 30°C.

For these aircraft models, the new engine(s) is (are) installed with no hardware change nor rating change.