By "direct", I mean that they stay in the same ship throughout the whole journey, though they stop at many different ports along the way.
Well, I suppose the best evidence on this topic is in fact a diary of the historical figure appeared in the game, Soseki Natsume. He was a very famous novelist in Japan later, and had studied abroad in London from 1900 to 1902.
Then, how popular was the ship route that Soseki left from Yokohama (near Tokyo) , and what kind of ship he boarded on Sep. 08, 1900 to Europe (Oct. 18 he arrived in Naples, Italy and then got off at the ship in Genoa on Oct. 19. From then, they left for Paris by getting on the train at Turin)?
The first part of OP's question, yes, he primarily didn't change his ship during the Empire Route between Japan and Europe, though his group sometimes boarded on a smaller ship to harbors the ship dropped in for a day or two-days short trip. The following is the list of such harbors en route:
So, the ship sometimes did not anchor in the harbor itself, and waited offshore for passengers who boarded on the smaller ship carrying them to the harbor.
The ship Soseki boarded on was called Preussen, and owned by German shipping company, Norddeutschen Lloyd (NDL). There were 4 companies who operated the Empire Route between East Asia and Europe in 1900 mainly biweekly, and one of them was a Japanese (postal) company. Preussen was a regular passenger ship, though.
There was no particular reason for Soseki himself to prefer German ship to others (including Japanese one), but his traveling mates consisted of 5 university professors and military doctor. All of them were government-sponsored exchange students (including Soseki himself), and the destination of some of them were Germany so they might have chosen NDL's ship. The government only allowed them to buy 2nd class cabin ticket, and they were divided in two rooms: Soseki and Michitomo Totsuka (Military doctor: 1868-1910) shared the room, and other 3 professors were assigned in the neighboring room (without private toilet and showers in individual rooms).
NDL's Preussen was not the largest ship on the Empire route (or even among the ships owned by NDL) in 1900, but it could carry about 330 passengers (1st class: 99, 2nd class: 28, 3rd class: 202), and its loading capacity amounted to ca. 5600 tons (Cf. Yamada 2017: 36). On the other hand, two other NDL passenger ships that they operated on the Empire Route in 1900 (Kaiser Wilhelm der Grosse & König Albert) were much larger, over 10,000 tons with more than 2,000 passengers. The Japanese had sometimes used the Empire route since 1860s (Cf. Kibata 2019), so it is reasonable to suppose that not all Japanese passengers were occupied by the diplomats and officials by the end of the 19th century. In fact, An English weekly paper published in Japan on Sep. 15, 1900, mentioned Soseki and co. as 'noted passengers' in Preussen despite of their passenger status (2nd class) (Yamada 2017: 39). In other words, there must have been other Japanese passengers than 'noted'.
References:
(Edited): fixes typos (sorry).