Is the biblical 'Eternal Life' the same as so-called 'immortality'?

No.

In Greek, there are three distinct words to describe different types of life: "Zoe", "psuche", and "bios". The "Eternal Life" mentioned in the Bible refers to Zoe. "Immortality," as commonly understood, refers to "bios" (biological life), while the "undying consciousness" widely discussed on today's social media refers to "psuche" (the soul or psychological life). You can search these three Greek words online, you will find more details there.