I wouldn't say it's a coincidence, but do disagree a bit. IMNSHO, there are two much bigger root causes of the rise of "smart" TV.
First is the increase in number of TV stations. The Big 3 broadcast networks, by nature of what they are, all work with the "appeal to everyone" mentality for prime time shows. Smaller channels (again, by nature) place a bigger value on having a smaller number of dedicated fans. So, it's natural that newer, smaller networks would push for what we consider "smarter" TV. The 90s is when cable TV shifted from being new to firmly established, and when a large number of the syndication TV networks were coming to power in the US.
Buffy was a flagship of the newly founded WB syndication network, and was the success that lead to their focus on teen shows (Dawson's Creek, Felicity, Roswell, etc). Babylon 5 was a flagship of the new Prime Time Entertainment Network. X-Files was a flagship of Fox, when it was on it's rise and working hard to try and steal viewership from the Big 3. All these shows were considered a success for the smaller networks, but may not have been on the Big 3.
The second thing is the rise of non-immediate viewership. Part of the reason why earlier TV always "reboots" at the end of the episode is because if you missed watching a show, you just missed it. It was gone. There was no way to simply hunt down last week's Cagney and Lacey because you went to bed early. By the 90s, VCRs shifted from being a luxury to a common household appliance. You could tape an episode if you knew you would miss it, and could even buy the episodes later if you wanted. I even remember lending VHS copies of shows to people when they missed an episode.
Of course, the marketing of the shows worked with this as well. I remember the continuity episodes of the X-Files being advertised as "events". And after things like Dallas became mainstream, viewers were trained to expect important episodes at season openers, closers, and sweeps weeks more than they were in the 80s. And, obviously, the monster-of-the-week-with-benefits format is built around letting viewers miss less important episodes and focus on the plot-critical ones.
So, yeah, the internet helped the fanbase grow, but I don't think it was the biggest factor. See also: the correlation of TIVO with 24 and Netflix with Mad Men and Breaking Bad.