When combined, 路 (lù) + 由 (yóu) + 器 (qì) literally forms “route-via-device.” This is a perfect, modern-era transliteration of the function of a router: a device that directs traffic *via* specific *routes*.
While a router is a universal piece of technology, its role in modern China has a unique cultural weight.
The use of 路由器 is straightforward and mirrors its use in English, typically in contexts of technology and home life.
The connotation is always neutral and technical. The formality depends on the situation but is generally used in informal, daily conversation.