In modern conversation, “唐僧” is rarely used to discuss Buddhist history. Instead, it's almost exclusively used as a vivid metaphor.
The most common use is to describe someone who is being extremely long-winded, repetitive, and preachy. It's a colloquial term used among friends, family, and sometimes colleagues in informal settings.
People might say someone “has started 'Tang Seng' mode” (开启唐僧模式了 kāiqǐ Táng Sēng móshì le) when they begin a long lecture. This is common on social media and in everyday chat.