"They are my older brother and older sister."
Which is the other sentence?
We normally can omit 我的 when "my" can be reasonably implied. If you omit 我的 in this sentence, i.e. 他们是哥哥和姐姐, we have no clue whose brother and sister they are.
Having said that, some similar sentences will be acceptable, but not related to possessive.
they are (elder) brother and (younger) sister
they are (younger) brother and (elder) sister
Note that these sentences would not make sense if you have 哥哥和姊姊, or 弟弟和妹妹.
Depends on the context I suppose. I was asked to translate from English to Chinese - in which case that translation would be inaccurate.
I was asked to translate "older brother and older sister", so translating them to an ambiguous "brother and sister" would be an inaccurate translation. In a different context that specificity might not be needed, so you could relax the translation and use what you wrote.