Depending on our point of view, we offer different approaches to address(ing?) a same issue.
Should I use —ing in this context, and if yes, why?
Both are correct. Each exploits a different grammatical function of the word to.
Depending on our point of view, we offer different approaches to address the same issue.
Here, to means the same as in order to. It introduces an infinitive indicating the purpose of offering the different approaches. The word "to" relates directly to "offer", as in "We offer low interest rates to make it easy for people to repay the loans."
Depending on our point of view, we offer different approaches to addressing the same issue.
Here, addressing is a gerund, serving as the object of to, which here works as an ordinary preposition. "To addressing" modifies "approaches": it describes the approaches. It's similar to this sentence: "We offer different methods of addressing the same issue."
And actually, to in the first version can also be understood as a preposition introducing a phrase that modifies "approaches". It's meaningful to speak of "an approach to address an issue"; it means the same as "an approach whose purpose is to address an issue". There are two ways to parse the first version, they both make sense, and their meanings are so similar that few people would even notice the ambiguity.
Yes you can, because Longman Dictionary says so, see the first example.
A noun phrase can be put after "approach to".