It's an alternate timeline, yeah.
In this one, the year is 2032. The world started to run out of mana before the turn of the millennium; putting most magi out of business. (The alchemists at Atlas are sort of getting by).
It turns out there's an alien supercomputer cube thing called Moon Cell Automaton, which has been observing Earth for unknown reasons for thousands of years; it has activated to create a pocket cyber-realm on Earth itself. (The records for each Servant are drawn from MCA's archives.)
This time around, the war has not 7 contestants, but 128. However, it's fought in a series of elimination rounds; so you as a player only fight about 7 Servants (the odd 'extra' notwithstanding, if you pardon the pun); one per week.
Also, apparently the Masters' souls get digitised and put into an avatar (either the male or female protagonist, in the case of the player); said form does not necessarily match what the 'real' person looks like.
(For example, the Rin in this game is not the same one from F/SN; Tokiomi had a liaison with someone while away from Japan. I'm not sure if this Rin is his daughter, grand-daughter, or some other form of descendant. She is only made to look like 'our' Rin by the Grail system.)
I'm on day 3 of week 7 right now, myself.