When did the word "home" become synonymous with "house", in contrast to an apartment home or condo?
This answer addresses mostly the use of the word home as a synonym for house, and just in the United States. I do not yet have a real answer, only a plausible possibility, for when the use of home started or became widespread.
My source is a chapter in the book Class; A Guide Through the American Status System by Paul Fussell. The chapter is "Speak, That I May See Thee", which is the closest US analog I know of to the British U and non-U English popularized by Nancy Mitford. Both works -- Mitford's and Fussell's (published in 1983) -- may be somewhat out of date now that money has come to be of such overwhelming importance, but I can testify, as a US native of too many decades (BosWash Corridor), that Fussell rings true on his discussion of home vs house. Fussel is a curmudgeonly sort, and something of a snob, but he got it right on home.
All classes except sometimes the upper-middle are implicated in the scandal of saying home when they mean house. "[Example]....they live in a lovely five-hundred-thousand-dollar home." We can trace, I think, the stages by which house disappeared.... First, home was offered by the real-estate business as a way of warming the product [as not] a passel of bricks, wallboard and Formica but snuggly warmth, comfort, and love......[several modest reasons why home caught on, followed by the hilarious]...(3) [to] the middle-class ...... house carried bad associations. One spoke of a rest home, but of a bawdy, whore-, fancy, or sporting house.
I don't know about #3, but it was too good not to include. But note that the bordello owner, Polly Adler, wrote her very popular book A House is Not a Home in 1953, which is about the time I postulate that home replaced house -- see paragraph below.
This paragraph suggests that home became popular in the great post-WWII house-building boom of the late forties and the fifties, when restrictions on using building materials for non-war purposes were eliminated, the economy boomed, and suburbs spread over the land.
As the comments have mentioned, the word home has not become fully synonymous with "house" in general. The meanings are similar, and some people may sometimes use "home" to mean "house" in the sense of "a separate building that serves as a residence for one household", but it isn't a usage that has become firmly established in current speech across the English-speaking world. (I don't know whether there are any specific regions where this usage has come to be usual.)
I doubt that this usage is particularly recent. More specifically, I'm sure that it is possible to find examples of people using "home" in the sense of "house" before anyone currently alive was born, so I'd expect it is "something I've just not noticed most of my life": an example of a recency illusion.
The Oxford English Dictionary entry for the the word home has an example of "homes" being used to mean "houses" from 1882:
Harper's Mag. Dec. 58/1 A lovely drive..is bordered with homes, many of which make pretensions to much more than comfort.
When I googled this quotation, I found a discussion of this topic that says that "seven years later" (i.e. in 1889), there was a real estate advertisement in the Kansas City Times and Star reading "for rent, a fine home at 1223 Broadway" (p. 71, "What You Mean by Home", in The Work of Poetry, by John Hollander).