As a point of comparison, Spotify, which counts 30 million paid subscribers to Tidal’s three million global users, said its best week of streaming belongs to Mr. Bieber’s latest album, “Purpose,” which debuted to 205 million global streams in seven days, and 224 million in its first 10 — fewer than Tidal’s count for “Pablo.” (“Purpose,” however, was not exclusive to Spotify, meaning it could also be found on iTunes, YouTube or even in record stores.)

Other top albums on Spotify have included Rihanna’s “Anti,” which had 309 million streams around the world in its first six weeks on the service after debuting exclusively on Tidal for a week. (A corporate sponsor also offered one million free downloads.) “Beauty Behind the Madness,” by the Weeknd, earned 311 million streams in its first six weeks, Spotify said.

By claiming to have topped his contemporaries by using a smaller platform and tighter exclusivity, Mr. West is declaring victory for his nontraditional release strategy. “This is not regular!” he tweeted of the numbers. (The industry standard counts 1,500 streams as equivalent to one album sold; 400 million streams for “Pablo” would equal about 267,000 albums, making it the best-selling hip-hop release of the year so far.)

Yet even in an industry intent on exploring corporate partnerships and selective streaming as alternative business plans while sales plummet, the more established channels eventually require attention.

Having given a free rein to Mr. West’s creative whims for more than a month, wider distribution by Universal proved inevitable. On Monday Def Jam began promoting the song “Famous,” featuring Rihanna, as a radio single, while also making it available to stream on Spotify, Apple Music, Google Play, SoundCloud Go and more.

On Thursday Def Jam said that “a newly updated, remixed and remastered version” of “The Life of Pablo” would be streaming on all major services by midnight on Friday, despite Mr. West’s earlier promise to keep it a Tidal exclusive. (“My album will never never never be on Apple. And it will never be for sale … You can only get it on Tidal,” he tweeted in February.) And with those services reporting plays to Nielsen, the album could make its Billboard chart debut two months after it was first heard.