No, an external HDD is a seperate HDD so you can install programs and save files and data completely independanlty to an internal HDD or another external HDD or any single PC or Laptop you use. Most external HDD's are plug and play so you just plug them into your PC/Laptop and it will appear as a seperate drive. If you unplug it from one computer and plug it into another the second computer will access the external HDD in just the same way as the first.
You might want to consider a wireless HDD or NAS, they might cost a bit more but any computer on your network will be able to acess it wirelessly. Some you can access from anywhere via Cloud, Browsers a dedicated app.
I have one of these:
https://www.amazon.co.uk/dp/B00FOKN7D8/?tag=gtplanetuk-20
As well as a Asustor AS-202TE NAS with 2 2TB HDD's:
https://www.amazon.co.uk/dp/B00FAKYCFO/?tag=gtplanetuk-20
Both work a charm, the Asustor has far more functionality and is used as amedia server plugged into a my TV, the My Cloud is used more for general data, photo albums and mp3's etc. I can access them from any computer on my network, or any device with the right app installed. Both are set up so they require a username and password so not just any old Joe can acess them.
I tend to use my external HDD and NAS as extra drives, I choose to duplicate some files across them but not everything. Games and things like that I can re-download if needed. Photo's and that kind of thing I tend to have a couple of copies of becuase I don't keep the original on my phone/camera. If you plan on doing a full backup, that should still work on any computer.
