The page you linked is from 2016 and is on the Ministry of Foreign Affairs (MOFA) site, with content specifically about overseas development assistance (ODA). I've read elsewhere on the Japanese web that the Ministry put out a statement that the page was not vetted for historical accuracy, and was intended to promote good-will relations between Japan and Mozambique. Just now, I found this 2023 PDF on the MOFA site (in Japanese) that mentions Yasuke as one of the early instances of people from Africa visiting Japan, wherein they make no mention of samurai status, and they even explicitly include a footnote stating that the Ministry makes no claims as to Yasuke's origins or status.
About the word kashin, this word has a broad range of meanings during that period of time. Saying that Yasuke was a kashin does not specify anything about his specific role or status in Nobunaga's household organization. Perhaps more importantly, none of the original historical documents describe Yasuke as a kashin: the closest we get is a mention that Yasuke was granted a fuchi or "stipend", which was basically a household salary, paid to employees of both samurai and non-samurai status, including basic servants.
About fighting, we only have evidence that Yasuke was present and fought in the Honnou-ji Incident (Wikipedia article), where Oda Nobunaga was killed. That said, this was not a battle so much as an ambush, where anyone and everyone capable of wielding a weapon would likely have done so. Yasuke's documented participation in the Honnou-ji Incident tells us nothing about his status in the Nobunaga household. When Yasuke was surrounded by Akechi Mitsuhide's forces, he surrendered to them, handing over his sword. Yasuke's captives then brought him before Akechi to ask what to do with him. As recorded in Portuguese by the Jesuits at the time, Akechi stated that, since Yasuke was a beast that knew nothing, and was not Japanese, he should be spared his life, and sent back to the Jesuits ("esse cafre he bestial, que não sabe nada, nem he Iapão, não no matem, la o depositem na igreja dos padres da India", letter from Luís Fróis of the Jesuits about the death of Nobunaga, dated 1582-11-05). Even had Nobunaga granted Yasuke samurai status at some undocumented point, it is clear that other samurai did not view Yasuke as a samurai.
About the swords, we again have no firm clarity in the original historical documents. These mention that Nobunaga specifically gifted Yasuke with a sayamaki. This literally means "scabbard-winding", and refers to the style of the scabbard. We don't know if this was an older-style tachi (traditional Japanese single-edged longsword) for which this scabbard style was first developed (with the winding added to protect the lacquer from wearing, as the tachi was suspended from the obi or sash and would rub against the wearer's thigh), or alternatively a more ceremonial short-sword with a fancy scabbard and no hilt-guard (for example, as described in Japanese here at Kotobank, a Japanese resource aggregator). We also have nothing in original documents to indicate a pair. Moreover, the whole "two swords = samurai" as a defining qualification doesn't happen until later in the Tokugawa period, so the number of swords isn't all that indicative anyway.
The key issue is that, going by the original documents in Japanese and Portuguese, we have nothing stating that he was granted the rank of "samurai". The lack of any such mention could be seen as telling, but then again lack of evidence is not the same as evidence of lack. We just have a lot of lacking evidence.
That said, given what has been documented, we also don't have clarity that Yasuke ticked all those checkboxes. He might have, but then again he might not have. We just don't know. Circumstantially, there are arguments to be made either way. Evidentially, we don't have enough information to say.
(FWIW, I vetted in "Preview", and offer my apologies for any weird misspellings caused by non-ASCII letters disappearing when I actually post this. I continue to be disappointed in the idiocy of Slashcode not supporting Unicode. FFS.)