In summary: Depending on how it is connected, it could act as a current source, a voltage source, or something in between. It all depends on the specific circuit and how it is designed.
How to recognise a constant voltage source? The volts are the same for a big range of load resistance values. i.e. you have a low (enough) source resistance. What does "same" mean? It means small enough not to matter in the context that the source is used. Most electronic measurements are 'good enough' if they are within 1% (more than enough for many purposes). If the source resistance is 1/100 of the load resistance, you are in the right ball park. (And electronics is very often a 'ball park' topic) It would be adequate for charging many types of battery at constant voltage, for instance.
You can apply precisely the same sort of criterion to determine whether you have a constant current source. Does the current vary significantly over the range of load resistances you plan to use? If you are after 1% accuracy, a high enough voltage source and a series resistor 100 times that of the load will also give you a 'useful' current source.
Those are examples of "rules of thumb". On some occasions, they would be entirely inadequate, of course.
Once you include some gain and feedback, your accuracy can easily be much better and you wouldn't actually need a vast voltage and high series resistance.