"-해서" is the shortened form of "-하여서".
Almost any verb ending in the form of "-하여서" is frequently shortened in the same way.
So we know "구별해서" is same as "구별하여서", which is then interpreted as "구별하다" + "-여서".
Standard Korean Language Dictionary has three meanings for "-여서".
The best fit for here would be "수단이나 방식을 나타내는 연결 어미", translated in National Institute of Korean Language's Korean-English Learners' Dictionary as "a connective ending used to indicate ways or means".
With the interpretation, the phrase "구별해서 부르다" is "부르다" with the specific ways of "부르다" by "구별해서": distinctly.
So your translation is pretty much precise, although I would suggest "How did they call distinctly each of you?".
- I feel the implicit object is more likely to be each of the two with a name, rather than just A.
- The role of "-여서" is closer to restricting the ways than the purpose. If it is "구별하려고" instead of "구별해서", "to distinguish" would be correct.
The Learner's dictionary contains some other examples of "-여서" with the same use.
- 살균 처리해서 유통하다 (distribute with sterilization)
- 꽃과 풍선으로 장식해서 꾸미다 (decorate with flowers and balloons)
- ("장식하다" and "꾸미다" is pretty much redundant .)
- 면접에 합격해서 회사에 입사하다 (enter the company by passing the interview)
- 반복하여서 읽고 따라하다 (read and follow repetitively)
As I translated the third example, the general correspondence in English for putting "-여서" to a verb is turning it into a participle.
It covers the all three uses of "-여서": sequence, reason or cause, ways or means.
Then, each use may be specified by adding a preposition "after", "because of", "by".
Usually, there is an English adverb to naturally replace the participle phrase.
Note that depending on the original verb, "-아서" or "-어서" should be used instead of "-여서", as in "보아서" and "들어서".
Also note that "-여서", "-아서", "-어서" can be shortened to "-여", "-아", "-어", so that "구별해서" is further shortened as "구별해".