This is a great question. Manuka is a common name for plants of the Leptospermum genus. As happens with many names, they can be adopted across many languages.
This is exactly what has happened with Manuka. It’s origins are clearly Māori. It was adopted as a common name for Leptospermum Scoparium. L. Scoparium is native to Australia and New Zealand.
Europeans from both countries started using the same common name for the plant. Honey varieties generally take their name from the common name of the plant.
Yellow Box, Ironbark and Leatherwood honey are a few common examples. This is also exactly what has happened with Manuka Honey. These days when people think of Manuka honey they are generally referring to honey from plants of the Leptospermum genus which possess MGO (methylglyoxal) levels in excess of 82. It is this elevated level of MGO which gives Manuka honey its unique healing properties.
Interestingly, the first recorded evidence of the name Manuka being used to describe leptospermum plants appears in a Tasmanian Government House of Assembly report in 1882. This is in Tasmania, Australia, not New Zealand. Shortly after, 1884 Tasmanian maps refer to areas of “low Manuka and Tea Tree scrub”, and the Launceston Examiner newspaper in 1884–1885 also describes Tasmanian landscapes of “manuka” plants