Is it also allowed for those who have Blue-Card issued from Germany?
This is probably an expats.SE question. Also, generally, the Luxembourg consulate is the right place to ask.
Working permits are still national matters in the EU and your German working permit does not allow you to work in Luxembourg. To the extent that your internship counts as work, you would therefore need some form of authorization from Luxembourg.
For less than 90 days, you can however travel to and stay in Luxembourg on the basis of your German residence permit and would only need a permit to work. For more than 90 days, even staying without working would be forbidden and you would need a Luxembourg residence permit/visa anyway.
A German blue card is a type of residence permit and as such also allows the holder to stay in another Schengen country under the conditions I just described. On top of that, there are special rules to make it easier to take up employment in another participating country but no unconditional right to work without applying for a local blue card.
Also not that a student internship in Germany will hardly yield enough compensation to get to the Blue Card threshold. Not even the lower one for IT. And moving the Blue Card from one country to other only works easily after some time, not immediately after one started working with it in Germany.
The better thing would be to make yourself available in LU for a period of 90 days and then within the timeframe get the appointment with LU immigration and change the blue card status to LU. Hope this works for you.
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