What is the purpose of the 中 in 在新标签页中打开?
在 and 中 are serving two different functions here.

The preposition 在 at the start of the sentence indicates where something happens. Meanwhile, the 中 here is technically a noun meaning "the inside", not "middle". It signifies that the link is opened on a new tab page, i.e. *the inside of a tab page".

Taken as a whole, then, the Chinese sentence is actually saying: Open link (打开) at [the position of] (在) a new tab page's (新标签页) insides (中) .

Without the 中 the sentence would still make enough sense to be understood, but it would sound weird. Generally speaking, when the target of 在 is not a location, it is standard to add 中 or 裡 etc. For example, 住在美國 and 住在公寓裡 .