"And" and "or" often have exactly the same meaning in English as they do in math, when they are used in the appropriate construct, however unlike math, they also have other uses and meanings.
Do you have a passport and an airplane ticket?
Clearly "and" is used the same as in math.
If you have a Red Carpet Club card or a first class ticket, you may board the plane first.
Here again, "or" is used in the same manner as in math. However, in English it is also used in other ways too, such as the examples you cite; English is not a precise in its meaning as math.
There is one particular ambiguity here worth mentioning.
Is that dog a Collie or a German Shepard?
Here there is a peculiar ambiguity. If the dog is a Labrador, the answer would be "no", but if it is a Collie, the answer would most likely be "a Collie"; to answer "yes" in this case would be considered pedantic. Math is, however, pedantic in the extreme.
This is odd because the question means something different depending on the answer. If it is a Labrador the question means "Does the breed of dog occur on this list?", if it is a Collie the question means "Please select the breed of that dog from the following list..."
Which is very odd, don't you think?