Is anyone allowed to call another person an apostate?
In case context matters, the person calling someone an apostate is a Pakistani Salafi.
Anybody can call anyone else an apostate. The real question is, who is taken seriously when that is done? In general, this is the job of a scholar, not the layman. Someone trained in the sciences of the religion to a high level.
Scholars are extremely wary of specific assertions of the status of someone's faith, simply because it is almost impossible to know what is in someone's heart. Both early and contemporary scholars therefore almost never issued rulings of apostasy or disbelief and erred tremendously on the side of caution (i.e. not issuing such a ruling). The only exception is when the outward manifestation of someone's creed is so blatantly obvious AND they pose a threat to the societal order that someone may reluctantly, for a specific purpose, issue such a statement.
The reason this is done so reluctantly is the statement of the Prophet (saws):
When a man calls his brother an unbeliever, it returns (at least) to one of them. Source
So for fear of the label returning to themselves, most scholars are extremely reluctant to do so, and only do so in cases where there is obvious and imminent harm. Even in those cases, as far as possible, it's a specific class of actions or people that the statement is directed against, and not specific individuals. For a scholar to issue such a statement directed at a specific person would mean that there were extraordinary circumstances indeed.
If you see someone doing this willy-nilly, painting broad groups of people with this brush, know that they are not serious about their scholarship and are unaware of the gravity of the matter, and are playing around with the rules of Islam. It is likely that there is an ulterior motive.