It doesn't need you to be logged in on any phone however. You can use just the desktop app. In fact I am on some really crowded and hyperactive Telegram groups so can't even imagine keeping it on my phone. Besides not a single one of my IRL friends use Telegram. I see 3-4 names, maybe they had signed up once.
It is for Whatsapp. Not only for sign-up, but actually all communication on the web version goes through the app running on your phone. The web "session" times out constantly, so I'd have to re-pair it with my phone all the time. If I still decided to use whatsapp.
Source: I did this for a few months. I'm not quite sure why...
How on earth does that work? Say my phone's on data, my laptop's on my home wifi with NAT. How does Whatsapp on the web reach my phone?
Threema doesn't require a phone number but it's not free.
-  https://hitta.se
Perhaps one of the virtual SIM card offerings would work with Signal though.
I'm juggling too many SIM cards and phones already. I don't want any new dependencies that complicate matters further.
I cannot ever let go of the phone number I used to activate Signal. In practice, that means I need to keep paying for a phone tariff I may no longer want.
It is true though that you do need to maintain a phone number. That is one of the trade-offs they made to allow it to actually be usable. I have not seen a better solution for encrypted communication yet.
What's wrong with the way Threema does it for instance, which is essentially to bind the identity to a key pair that you can back up?
[Edit] And I think it's essentially the same with WhatsApp, only they now make it a less manual process.
No smartphone is required for JMP, only an XMPP client (see https://news.ycombinator.com/item?id=16352711 for some client options).
All it require is a phone number, besides it's Snowden approved :)