Bob's job at Insuricare involved him working with a desktop terminal/computer Given the limits of technology, it seems likely that such a device would have a screensaver to prevent the classic green screen burn-in .
You might also want to note that the concept of a screen slave (in the literal sense of a terminal slaved to a mainframe) has been around since the 1950s .
The metaphor didn't, I assume, go unnoticed by those who had to sit in front of them all day for work, nor by those who spotted that their friends and family were entranced by the idiot box in the corner of their living room.
“Screens were hypnotizing us 30 years ago,” he explained toHelloGiggles.
“Where everybody was plunked down in front of thetelevision and just changing channels and staying there all day.
Theidea of people kind of forgetting what's going on around them, infavor of what's going on inside this cool glowing box is not a newidea, but it is one that stays with us.”
The new villain in Incredibles 2 makes the movie sound like a terrifying episode of Black Mirror
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Because the film isn't actually set in the past While the film has the appearance of something out of the 60s, it's actually set in a stylized present, imagined through the lens of what the people in the 1960s thought the future would look like.
This is made more confusing by the fact that Bob reads a newspaper dated May 16, 1962 in the first film.
But in a Bustle interview , writer/director Brad Bird and producers
Nicole Grindle and John Walker explained: "I probably wouldn’t have put 1962 on the newspaper if I thought it would get out," Bird said.
"I didn’t even know that we did that.
I just wanted it to feel like the '60s.
" Grindle added, "It’s more just like an aesthetic choice than anything," while Walker went a step further, saying the idea was to create a future based on the 1960s. "You [Brad] always said it’s like the imagined future from the '60s," Walker said.
"That’s when it’s set.
What people in the '60s thought was coming."
(emphasis mine)
This isn't a horribly uncommon style decision either, especially in the superhero genre.
Retrofuturism
As @CaptainMan mentions in the comments, the idea of an imagined future from the vantage point of a specific time period is a style known as retrofuturism and is the basis for the stylized realities seen in shows like Batman:
The Animated Series (and Bruce Timm's other DC Animated Universe shows) as well as the steampunk genre.
From the Wikipedia article: Retrofuturism (adjective retrofuturistic or retrofuture) is a movement in the creative arts showing the influence of depictions of the future produced in an earlier era.
If futurism is sometimes called a 'science' bent on anticipating what will come, retrofuturism is the remembering of that anticipation.
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Screensavers as we know them didn't exist then.
I don't recall the character admitting that it was a pun, so I assume that the name comes from the behaviour.
Screenslaver enslaves and ensnares people through the medium of the screen - especially those who are already victims of screen addition.
The quote given in Valorum's answer describes this.
The "Screenslaver" name is a pun primarily intended for the (modern) audience, but it does describe the nature of the crime involved.
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I was thinking screen enslaver got morphed into screenslaver, as the looking at screens was how folks got hypnotized/enslaved.
Not necessarily a screen saver.
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You missed the point "screenslaver" isn't necessarily a pun on screensaver
it's a pun on consumerism of media and radio in a broader sense.
For 30 years of its existence (1936–1967) television was entirely in black and white.
CRTs were also invented before the 1930s with the first commercially made electronic television sets with cathode-ray tubes were manufactured by Telefunken in Germany in 1934.
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The Incredibles  2, the Screenslaver's name is a clear pun on "screensaver".  Screensavers were invented in 1984.  The film takes place in a broad 1950s pastiche, and we don't see any personal computers or other similar monitor based devices, rendering this pun completely useless.  Yet a villain as obsessed with invention and design as The Screenslaver would surely not pick a name that's completely meaningless! What, in-universe, could have prompted them to name themselves this? What is the joke, if not on Screensaver?
In The Incredibles 2, why does Screenslaver's name use a pun on something that doesn't exist in the 1950s pastiche?
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The question is incorrect to state that personal computers don't exist in the film, as we see Bob working on a computer terminal at work. Furthermore, whilst the word "screenslaver" is a pun for the audience, it also describes the crime being committed, enslaving people through a screen. Additionally, it should be noted that the concept of a "screen slave" has been around since the 1950s and that the film is actually based in the stylised present. Views vary - Bob might have had a screen saver even in the 1960s, but maybe the film was not even set in the 60s but in the future as imagined in the 60s and maybe 'screenslaver' is intended as a broader pun on the consumerism of radio and TV, with resonance at least for today's audience. 
Bob might have had a screen saver on his computer at work even back then, and the idea of people being enslaved to the TV was also current. The film is not actually set in the 1960s - it is set in the world of the future as imagined by people in the 1960s. Screensavers did not exist back then but the term 'Screenslaver' has a dual function in that it describes the nature of the crime whilst operating at the same time as a pun for the audience of today. Screen enslaver got morphed into Screenslaver - it does not have anything to do with screen savers. The term "screenslaver" isn't necessarily a pun on screensaver - it is a pun on the consumerism of media and the radio in a more general sense.