There are different things in mathematics that are called space: Topological space, vector space, etc. Wikipedia defines space as below:

And in the page related to the word "structure", algebraic structures are listed too. So why people never use the word "space" when talking about groups, fields, rings, etc.?
It seems people tend to use this word for things that have a more geometrical nature. Or is there something else?
I'm asking this because rigorous mathematical texts don't do it too so there is a chance they've got a reason for it!
Thanks

Not sure why but its always better to use different words for different things instead of overloading a word too much. When I think of space I think of the mathematical combination of multiple identical fields along with some sort of metric like Pythagorean theorem which I guess is the added structural component.

The notion of a module might complete your understanding and tie things together better:

I always think of "space" as a framework in which things exist. So "vector space" makes perfect sense and this follows the definition you gave exactly except that I'm not rigorous in what I'm calling "a set". I don't know if it would make sense or not to say a "group space" or a "ring space" since I don't know those areas of math.