-The Japanese did not think Portugal was the only country in Europe. In fact in 1582 a Japanese mission was sent to Europe representing the Christianized southern Daimyos who wished to establish formal relations with the Catholic church. Members of the mission were ordained Jesuit priests and returned to Japan in 1590 (10 years before Shogun is set). Japan's history as 'isolationist' while not untrue has often been greatly exaggerated in the West. It would be more accurate to say the Shogunate strictly controlled contact with the outside world and mainland Japan, rather than outright banning it but the late 16th century was a period where regional lords had more control over their own foreign relations before the Tokugawa centralized power.
-The katana, while it existed in this time, would not have been a primary weapon or symbol of Samurai class at this time. The weapon's ubiquitous adoption owes heavily to Edo era sword laws enacted by the Tokugawa shogunate.
-The Japanese military at this time, is way more modern than you probably imagine. In 1600 the Imjin War in Korea had only just ended and was fought by Japanese armies that had largely converted to firearms. This wouldn't change until after the Battle of Osaka (the last battle of the Sengoku era in its entirety) when internal warfare in japan essentially ceased and there simply wasn't much use for large armies armed with guns anymore. In 1600 though, a Japanese army would have had thousands of guns.
-Historically, by this point Tokugawa Ieyasu was gearing up for the Battle of Sekigahara, which essentially secured his rule and the period was mostly over save for the last holdouts plotting top elevate the heir of Toyotomi Hideyoshi.
-European trade with Japan had a history of being very regulated, as was the work of missionaries. Tokugawa largely reasserted these policies and continued them after a brief period of pause where Toyotomi Hideyoshi had more open relations before he became paranoid Christians were conspiring to take over the country. While the series makes it seem like the Portuguese had a monopoly, this is only loosely true. Portugal got to Japan first by several decades but eventually the Dutch, Spanish, and English were also trading with Japan once they got there through controlled ports like Nagasaki, but also on the southern islands of Okinawa and Ryukyu islands where Japanese merchants could trade more directly with Europeans outside of the Shogunate's direct authority and then bring the goods back to Japan.
-Tokugawa's historical interest in Adams was mainly his knowledge of Western shipbuilding and trade relations with the Netherlands. He arrived a few months before Sekigahara and established the first trade partnership between Japan and the Dutch and helped get Tokugawa in touch with other European traders.

If I remember correctly, Anjin was named Hatamoto after saving Toranaga during an earthquake in the old series but in the modern one he is named Hatamoto after helping him escape from Osaka.