I have applied to a PhD student position via the lab portal (as requested) and my application statut is "Application under review" for more than month now. Is appropriate to contact the PhD supervisor directly via his email and ask for news.
Actually, I'd suggest this only under certain circumstances. But if it just anxiety, I'd recommend waiting.
But if you have decisions to make or if the deadlines are fast approaching it would be appropriate. Likewise if you have additional information that might affect the decision.
If you know that the decision will be made by a certain date then you should learn by then in any case. But if it is clearly open ended then it seems more appropriate to both state your continued interest and ask for any information. And a month or so, in that case, is probably fine.
And, of course, don't close out other options while you wait.