Is there just one EM field for the whole universe? Does our universe contain individual magnetic fields ?
For example two different magnets, one here on earth and one on mars. Do both of them have their own magnetic field? Or is there only one single field that stretches across the entire universe, but have different strength on different locations? For example around a magnet ?
 A: Yes, there is one universal electromagnetic field
Clear answer: yes, there is only one electromagnetic field for the whole universe.
Ontological answer: You can go to any spot in the universe and take a measurement. Even if you were to find a large chunk of space with "0" as your result (and for this part of the answer we do not care if this is physically possible or not), the field itself would still be there, displaying those "0"'s. Oh, let's ignore black holes as that's obviously not the idea you had in mind.
Physical answer: If people talk about the "field of an electron", making it sound as if that were some kind of localized thing, then that is just a shortcut to reduce incredibly complex maths to something they can handle. None of the interactions have a hard border or a defined cut-off radius. That quite literally means that every single charged particle in the universe influences/interacts with every single other charged particle in the universe, no matter how far away it is. If you wiggle your finger here on earth you will definitely change the EM field on Mars. In practice this influence is so utterly, unimaginably small that it obviously is neither measurable nor anything you need to keep track off. But it is still there. 
Regarding the comment on this: Finger-Mars is obviously a half-joking example. The point is not the light cone, event horizon or the magnitude of the influence, but that it *is* the same *field*; the Finger-wiggling does propagate (at least in our theories), but obviously with such absurdly small numbers as to be totally irrelevant.
A: You can define a "wind field" for the Earth by putting a weather vane at every point. You've probably seen drawings of these wind fields in weather reports; you can even define 'wind field lines' in analogy with electric and magnetic field lines. Then a completely analogous question is, "is there just one wind field on the Earth, or does every storm have its own wind field?" 
Both descriptions are correct, but each might be more useful in different situations. If you want to know how breezy it'll feel standing in one spot, the total wind field is better. But if you want to get an idea of the global structure, you might want to decompose it into contributions from each storm.
The exact same reasoning holds for the electromagnetic field. You can say it's the sum of fields from each charge or current, or that there's just one total field. The underlying physics is the same, because electromagnetic fields obey the superposition principle. 
A: If there were many different electromagnetic fields, then we would have many different types of photons. Even if all these different fields and photons would interact with matter in exactly the same way, it would still have experimental consequences. In processes where photons are emitted, the probability of emission of photons would increase proportionally to the number of photon types, e.g. the Sun would radiate more power.
A: Short answer: Yes, there is only a single field whose value varies by position and time.
Long answer: There is only one kind of EM-field, because there is only one gauge boson for the electromagnetic force: the photon. And since photons represent the energy passing through the EM-field, fluctuations in the field propagate at the speed of light.
However, due to the rate of expansion of the universe, some regions of space are moving away from others faster than light. This means that some patches of space are not causally linked—particles in one patch of space can only contribute to the portion of the EM-field "trapped" in that bubble while the rest of the universe expands away. 
Regardless, the same laws of physics still hold true in all* patches of space in the universe, so there is still only "one" all-encompassing field. It just means that some components will be forever isolated from others, therefore they cannot interact with each other—through the electromagnetic force or otherwise.
*Actually, modern physics doesn't work if the energy or gravity exceeds a certain point, like in a black hole.
