As with training environments generally, the story presented is simplified to an extent appropriate to need. Stalling is no different.
For a given aerofoil... does the stalling angle change (provided flaps are up) of course...
Well, yes and no.
The basic training story ignores both Re (Reynolds Number) and M (Mach Number), both of which have a significant effect on CL .. but not to any significant extent at low speed and altitudes where basic training occurs in the civil environment.
Similarly, one needs to be cautious in the interpretation of the usual wing section CL properties typically seen in textbooks .. these are basic engineering starting points but relate to wind tunnel work with wing sections but not the whole aeroplane. Stall characteristics for the aeroplane will be influenced considerably by tail loads and fuselage effects.
Further, the typical POH stall speed relates to the relevant Design Standard certification stall .. an animal which doesn't get seen much in the real world albeit it is the starting point for a lot of speeds which we do use routinely in line operations. The real world stall is very much affected by pitch rate, yaw inputs, etc. The actual incidence angles achieved at stall can vary to some extent as a result. There is a particular phenomenon associated with very high pitch rates where the formation of a spanwise vortex sees the stall angle go to quite a high, if short-lived value ... not something you will see on the typical bugsmasher but, certainly, something of interest to the rotary wing folks.
Is it true in regards to larger jet aircraft?
Apart from Mach effects on CL, in particular, a stall is a stall and one sees a degree of consistency in the basic characteristics for a Type-Certificated aeroplane flown in a certification manner.
The graph CL over AOA is given to a specific aerofoil and thus aircraft type
The CL characteristic curve relates to the the wing section. A particular wing section may be seen on various aeroplanes. As indicated previously, once the wing section is incorporated into the whole aeroplane animal, things change a bit although the basics are still much the same, if not the specific numbers.
Is that graph true for cruising speed?
The typical section graph relates to very controlled, steady state, wind tunnel work. With that caveat, and sensible pitch rates, the main effect of speed will be to demand a different body angle to achieve the required CL for the particular stage of flight.
I was told because CL is affected by speed and Reynolds number the stalling angle cannot be fixed for a specific aerofoil thus it is variable...
Perhaps your story teller referred to Mach Number, rather than speed ? Two different things. If you are interested in Re and M effects, there is plenty of useful information on the net via the services of Mr Google et al. As indicated previously, for low altitude, low speed flight, Re and M have very little effect and can be ignored, as they are for light aircraft.
There is some more of the story in this thread -
www.bobtait.com.au/forum/rpl-ppl/5022-different-types-of-stalls