No, Zatoichi never fakes his blindness. He is truly blind. At least that is what most people familiar with the tale of Zatoichi and the series would tell you. The character is intended to present as a weak blind man who is strong on the inside.
The Zatoichi films, not only a 2003 movie but a long running series, were first made in the 1960s, with the main character being a seemingly harmless weakling, a pushover in appearance. A masseur by trade, he wanders the country. Masseurs, both the profession and the practitioners, were known as anma/按摩 in Japan. Most masseurs were blind. The character's name is actually Ichi (市) and zatō (座頭) is an old-fashioned way to call the social class of blind men working as masseurs, acupuncturists, and lutists.
The film series' Wikipedia page has pretty detailed explanation of the character:
The character's name is actually Ichi. Zatō is a title, the lowest of the four official ranks within the Tōdōza, the historical guild for blind men (thus, zato also designates a blind person in Japanese slang). Ichi is therefore properly called Zatō-no-Ichi ("Low-Ranking Blind Person Ichi", approximately), or Zatōichi for short. Massage was a traditional occupation for the blind (as their lack of sight removed the issue of gender), as was playing the biwa or, for blind women (goze), the shamisen. Being lesser Hinin (lit. "non-people"), blind people and masseurs were regarded as among the very lowest of the low in social class, other than Eta or outright criminals; they were generally considered wretches, beneath notice, no better than beggars or even the insane—especially during the Edo period—and it was also commonly thought that the blind were accursed, despicable, severely mentally disabled, deaf and sexually dangerous.
And actually your quoted lines are pretty telling of the character. When asked if he is blind, Zatoichi replies: "Not at all." He does not mean he is not physiologically blind. He is. No doubt about it. He is saying that he is not blind in the heart and with his heart he can see clearly all the darkness in the world that he is battling. In the same vein people say "a blind heart" or "blindness of the heart". It is for the same reason another character comments: "I knew that you could see! I knew the first time you came here." meaning: "I know you could see clearly with your heart. You are a sage."
Note: It has been a long while since I saw this movie, so I am writing from memory, but I think the gist is clear.
The greatness of the character of Zatoichi lies in the fact that he comes from the lowest rung of the society, he is severely physically challenged, and he is looked down upon by almost every new person he meets. But he is strong, both spiritually and in swordsmanship. Keywords: contradiction, surprises, spirituality.