Whether that makes it the "same word" is a philosophical question. But writing "hànzì" is proper when referring to the use of the characters to write Chinese. If one is using it to mean a set of characters (rather than the general concept of characters that come from that writing tradition), they're different sets; and there are typically different expectations for typesetting etc. The decision to produce "CJK Unified Ideographs" in Unicode was not without controversy, and quite a few words have been spent by standards committees on explaining why these characters should share code points while there are completely separate Latin, Greek and Cyrillic scripts (despite shared history and many at-least-seemingly overlapping glyphs).

The word “hanzi” in English is much less commonly used — people studying or discussing Chinese are more likely to call them “Chinese characters” or just “characters”.