While it has been mentioned already that Batman: The Animated Series and Mask of the Phantasm occur in the same universe (maybe), almost every source agrees that Phantasm is set in the 1940s.
The description of the film released with various official merchandise (e.g.) states:
In this 1993 animated feature set in the 1940s, Batman is pitted against a mysterious figure who is taking out Gotham’s most dangerous criminals and many believe Batman is responsible for the murders....
The 1940s claim is reinforced by various articles, both old and new. For example:
The DCAU Wiki notes the discrepancy between the time period of TAS and of Phantasm, pointing out the existence of sources that place Phantasm in the 1940s. (It also notes that the reason TAS is believed to take place in the 1990s is the appearance of a newspaper in one episode that has "1992" on the front page.)
How does one rectify this? Note that the answers to the "same universe" question are not the most conclusive. One answer establishes that the Phantasm is a character in the DC Animated Universe. All that means is that the Phantasm was "good enough" to be used again, in a possible different continuity. Another answer relies on a particular photo of the Joker appearing in both Phantasm and the Animated Universe.
Myself, I am not especially convinced that Phantasm occurs in the same universe as TAS. Rather, it could be an homage to TAS, meant to be viewed as an independent story.
As for the Bat-Computer, it is out of place in the 1940s, but:
Wayne might have had access to technology that the general public was not aware of. We do not see a computer anywhere else in the film — not on any desks or in the hospital scene, etc.
We might be seeing an alternate 1940s, with slightly more advanced technology than we ourselves had at the time.
In short, there are no direct references to the 1990s in the film and no sources stating that it takes places in the 1990s, but there are plenty of sites insisting that it takes place in the 1940s.
This is probably not what you wanted to hear!