As Wikipedia says, the actual Lake House was built from scratch for the movie and had to be deconstructed after filming. According to this article this was apparently part of the conditions for using Maple Lake in Willow Springs, Illinois as its location, to leave the place in the exact same condition they found it (though, they actually built a new fishing pier in its place instead).
According to the movie's production notes they had indeed searched for something existing at first, but
After weeks of scanning lakefront locations in Wisconsin, Michigan, Illinois and Indiana and virtually every mile of the vast Lake Michigan shoreline for something that embodied all the necessary elements, The Lake House production team rose to the challenge and built the iconic house themselves.
Especially since this way they could tailor it exactly to what they needed without making compromises. According to production designer Nathan Crowley
“The house was our most important set, the one key piece around which the story revolves, so it made sense for us to create exactly what we wanted.”
In fact it was production designer Nathan Crowley himself who designed it, as he actually acquired a B.A. in architecture from Brighton Art School before he delved into filmmaking. All in all it took only 10 weeks to finish the house beginning from its initial inception:
2 weeks design and documentation, 4 days bidding, and a scant seven weeks for actual construction.
The engineers bulding it were even nominated for the Structural Engineers Association of Illinois 2006 Excellence in Structural Engineering Awards for their efforts to construct that house under all the temporal and structural restrictions the filmmaking imposed on it.
As to how it connects to the movie, Crowley first and foremost wanted something that the in-universe designer, Alex's father Simon Wyler, could actually have designed himself. So they chose the modern arichtects of the 60s as a base inspiration (as an admittedly not very architecturally knowledgable person I'm most reminded of Mies van der Rohe with his simplistic steel-based constructions and his penchant for large glass areas). But they also added influences from earlier eras to spice it up (which to me seems noticable in the smaller scale and more playful and detailed aspects of the house). According to Crowley
“We wanted a mixture of modern and classical, something that Alex’s father might have designed. In some ways its roots lie in the kind of 1960s glass box style, but we also drew from the Regency Period, a style popular in England in the early to mid-1800s, to add romance and elegance to the overall look, and even integrated some elements of greenhouses.”
In fact the rather unique style of the house further helped to establish a connection between Alex and Kate, since they both like that house very much. According to Kate's actress Sandra Bullock that marks how close and similar these two actually are:
“It’s such an unusual design, undeniably beautiful but not the kind of design everyone would fall in love with. Only a certain kind of person could live in this house. It’s for very specific tastes. For both Kate and Alex to feel so comfortable within these glass walls shows how much they have in common and is symbolic, really, of the greater understanding they offer each other.”
She also yet again emphasises how the contrast between modern and classic adds to its impression, as it specifically reminds her of
“Paris Metro stations from the turn-of-the-century, clean and minimalist with lots of glass and steel. It’s a design that might sound cold but is actually quite warm when you see it.”
Apart from its architectural style, the house of course has an even deeper connection to Alex, as his father actually built it out of love for his wife and the family once lived there happily, before his father's ways estranged Alex's mother. According to his actor Keanu Reeves he seeks to get back those happier days by restoring this house after all this time:
“He’s trying to make this house a home again.”
Which also adds into his (creative) conflict with his father when Alex in the movie says:
Dad knew how to build a house, not a home... I think he wants us to do
what he couldn't.
But the house's specific design also plays into that, since as it stands so desolate in the middle of nowhere, "there are views all around, but no way to connect with them." As glass and open as it seems, the glass also separates you from the surroundings and there is actually nothing else personal around it.
And this desolation yet again adds to Alex's and Kate's rather introverted and lonely personalities in the film. The remote location separates them even more from the rest of their life and environment. But at the same time, the open glass structure also forces them to not go into complete hiding either but rather exposes their true nature. According to Sandra Bullock
“These are two people who are, in many ways, hiding from the world, and yet the house they both love is absolutely exposed,” as if the mere fact of their being there brings out their genuine nature.
So to sum up, the filmmakers had to build the house from scratch to serve the exact needs they had and production designer Nathan Crowley's education as architect enabled him do so. The design was primarily chosen to be realistic in terms of the films narrative, but at the same time unqiue enough for marking an interesting visual aspect and an implicit connection between its two inhabiting protagonists. And it's glass structure in the middle of nowhere provides enough conflict between being open yet remote to emphasize the character's introverted nature and their disconnection from their environment together with their mutual connection.