You can drop it anytime, in any gender, because the verb indicates the person and the number. When you leave it (Αυτό είναι ένα παιχνίδι), it sounds more emphatic and more like you're saying This is a toy.
I just realised... but παιδι is a child... So παιχνιδι is kinda like a child thing? Soz guys I speak multiple languages by finding these little connections which might or might not exist
παιχνίδι is from παίζω "to play", but that originally is related to παις "child", which is where παιδί comes from -- the original meaning, I suppose, was "to play like a child".
The same order and structure is in Italian, they also drop pronouns. Example: È un uomo. (Lui è un uomo). Είνσι ένας άντρας. (Αυτός είναι ένας άντρας).