The phrase in this day and age means at the present time.

It is a tautology, that is to say, the words day and age are synonymous, day having here the sense of a period of time.

The earliest instance that I have found is from Leisure Moments.—No. III., by ‘Ytus’, about the need for farmers to occupy their leisure moments in the study of agricultural papers and books, published in The Genesee Farmer and Gardener’s Journal (Rochester, New York) of Saturday 24^th March 1832:

In this day and age, no one who is blessed with health and a common share of that indispensable article, good sense, can have an excuse for ignorance, and that hyena monster with her numerous retinue scattering desolation over the earth, ought not to find a resting place in our land.

The second-earliest instance that I have found is from an article about the abolition of slavery in the District of Columbia, published in the Carlisle Herald and General Advertiser (Carlisle, Pennsylvania) of Wednesday 20^th January 1836:

It is too much, in this day and age, to hear Chief Magistrates, distinguished Senators and Legislative bodies attempt to advocate and justify slavery on the principles of ethical science and the christian [sic] religion! This we say is too much in this day and age. Such nonsense, such absurdity is not excusable in school boys, but it shows how, what are called great men, can talk and argue when influenced by prejudice and interest.

This Day and Age (1933) is the title of a film directed by Cecil Blount DeMille (1881-1959); the following review by Hubbard Keavy was published in the Tampa Morning Tribune (Tampa, Florida) of Sunday 30^th July 1933:

Boys Rout Gangsters In New Picture

Hollywood, July 29.—(Associated Press.)—One hears that gangster pictures are “washed up,” but along comes “This Day and Age” to disprove the statement. Cecil B. DeMille has put a new twist on a subject that Hollywood had just about run into the ground with sameness.
DeMille has entered two new fields, gangsterdom and the younger generation, which gives some hint of the story. Gangsters are pictured as cruel tyrants, in true movie fashion, but the juvenile element is given a clean bill of health, something new for “flaming youth” in motion picture circles.
The picture, obviously, is not offered with serious intent, but as someone’s fanciful idea of what might be done about the gangster menace.
The shock of the picture is the surprise twist, with all of the gangland activity subordinated to the running story of how a group of high school boys, filling governmental posts for a day during “boys’ week” observance, begin a fight that ultimately upsets the works for a whole gang of hoodlums.
The slight love interest introduces pleasing Judith Allen¹ (Mrs. Gus Sonnenberg) as the high school flapper who keeps the gangster’s bodyguard occupied while her boy friends engineer their coup.
Charles Bickford² is the gang lieutenant and Richard Cromwell³ the guiding spirit in the cleanup campaign, Eddie Nugent⁴, Ben Alexander⁵, Harry Green⁶, Bradley Page⁷ and a host of others contribute to the realism by good performances.

¹ Judith Allen (1911-96)
² Charles Ambrose Bickford (1891-1967)
³ Richard Cromwell (born LeRoy Melvin Radabaugh – 1910-60)
⁴ Edward Nugent (1904-95)
⁵ Nicholas Benton ‘Ben’ Alexander (1911-69)
⁶ Harry Green (1892-1958)
⁷ Bradley Page (1901-85)

The following advertisement for Burton tailoring was published in the Lancashire Evening Post (Preston, Lancashire) of Thursday 18^th October 1956:

In this day and age . . .
“Now there’s a spot of glamour for you, dear,” she says. “Must have been hard work keeping up with the Joneses in those days!”
He chuckles. “Me for the age I live in,” he replies, “I’m all for keeping up appearances, but I like things simple and straightforward” . . .
Yes, there’s the man of to-day talking. Probably it was just those reasons that brought him to Burton’s for the coat he’s wearing himself.