Yes, those day names were likely borrowed directly. However as the terms "sun" and "moon" themselves are also significant as matching with the concepts of yin and yang, that was probably quite intentionally done.
This happened at the same time that Japan adopted a calendar based on 7-day weeks. Other than Sunday and Monday, the days of the week are named for the elements which represent the 5 planets known at the time. They are fire (火, か ka), water(水, すい sui), wood (木, もく moku), gold (金, きん kin) and earth (土, ど do). These are Mars, Mercury, Jupiter, Venus and Saturn respectively.
Officially, Japan adopted the Western calendar in 1872 near the end of the year, with the change taking effect starting the following January 1. Before this, the calendar system used in Japan was based on that of China.
Edit: I'm pasting in follow up's here just to keep things in one place in case someone finds this later and is curious.
Did the seven-day week exist in Japan before the name change?
Not really, no.
It was something people knew about as far back as the 800s but it wasn't something that had any meaning to the average person. Prior to this change, the year was divided into 12 months, which each consisted of 30 or 29 days. Each month was then divided into three 10-day "weeks", with the last day of each "week" being a day of rest. These periods are called tōka (等価), though that also refers to the tenth day of each period/week.
The name change brought with it a new solar calendar replacing the traditional lunar one and with that new calendar came a whole new way of dividing up time.
When did China switch to the 7-day system?
China's a little complicated because multiple systems existed at the same time. Like Japan they knew of the idea of a seven day week long ago. In China's case it was during the Tang Dynasty in the 600s, and for Japan it was only a couple hundred years later. But it didn't really serve much purpose to the general public and so it wasn't the main way of organising your month.
Officially, Republican China adopted the Gregorian calendar n 1912, to include and with that the 7 day work week. This was the year after the founding of the Republic of China. In Taiwan (the Republic of China) the years are numbered differently, and it's now year number 104 because you count starting with 1911. Still they recognise the Christian calendar year and use it for a lot of stuff.
Before that China used a 10-day week like Japan, but in Mandarin it is called a xún (旬). They knew about a seven day week through contact with religious missionaries, but didn't officially use it until 1912. The calendar actually didn't have more total official implementation until the end of the 1920s, since before that point the Republican government didn't really have as much control of the country.
Communist China was founded in 1949 and by then the Western calendar was already the norm, an that's what they used from day one.
However China has historically had multiple calendars that were in use, and today you'll still find what's called the "farmer's calendar" (農曆 nónglì) which is the traditional lunar calendar. This is usually the traditional day numbering printed in addition to what you'd expect of a Western calendar These different systems served different purposes, and they were aware of other ways of calculating the days well before any official adoption of the Western calendar.
Did Taiwan ever operate on the 29-30 day lunar system?
Yes, but in the limited context of this farmer's calendar. It's not the calendar that the banks run on or anything like that. It is more of a traditional system that the majority of Taiwanese people pay no attention to.
Sources:
hi! you'll find a bit more info on this here