Oh defiantly not - White Star Line is the premier brand of the group; you'd never use that for plain cargo. Leyland is the most likely I'd have thought.
What happened to P&O, Blue Funnel and some of the other lines?
P&O is still going strong. White Star attempted a buy out in the 1920s, but was unsuccessful. Behind White Star, they are the second largest passenger line in the UK, and one of the largest in the world. They focus primarily on shipping to India, Australia, China, Japan, etc. They have only limited services to North America and no true superliners like White Star does. Their ships are primarily in the 500-800' range. Unlike White Star, they operate a pure cargo operation as well, though it is fairly limited with their focus being on passenger shipping.
Blue Funnel is also still kicking, though strictly as a Cargo line. Containerization has occurred roughly as IOTL. Not quite as fast, but only 1-3 years behind due to a shorter war in the 40s.
A lot of the smaller, independent lines have either merged with larger companies or give out of business as they couldn't complete with the big boys any longer. With all of the major lines operating ships in the 80,000+ ton range, they just couldn't offer comparable levels of comfort or price. A few tried to develop cruising as a way to save the company, but their ships, like almost all liners, were distinctly unsuited for it.
Some of the lines, like the Greek Line are still around. Though operating smaller new build ships or larger secondhand ones. All in all, passenger lines are still a thriving business. Though as ships get ever larger, the smaller lines are slowly disappearing as they can't afford the big ships.