How it works:
1. Phone: Start BankID on your phone. The program will say that it's waiting for a connection.
2. Computer: You go to your bank/government service web page, select Mobile BankID, then type in your 'person nummer' (like SSN).
3. Phone: BankID on your phone will then prompt you for your PIN.
4. Computer: You're logged in.
It's all happening over the internet. Sure, it might not work if you change SIM, for example, but at least it's not telephony based.
Best of all, I could get an activation code for mobile bankid using the win 7 bankid instance I set up in http://verahill.blogspot.com.au/2014/11/602-surviving-bankid-rant-moving-nge.html and now I never have to use it again.
[if you're on linux and in Scandinavia, just install mobile bankid on your phone or on android in a vm]
So, if you're on linux and you've been using the official bankid application and you happen to be a couple of continents removed from Scandinavia follow this post which basically does the following:
1. Install ubuntu 10.04 LTS in a VM
2. Install bankid 4.19.XXXX or earlier in the ubuntu VM
3. Copy your ~/.personal from your regular linux computer to the ubuntu VM
4. export your key using persadm export to a USB stick
5. Install windows 7 in a VM using the free, legal isos and the free, legal installation key. The copy will expire after 30 days (but will still continue working).
6. Install bankid in the Win 7 VM
7. Under File/Preferences in bankid add the directory on your USB stick with the exported bankid key
8. Log in to your BankID issuer (probably your bank) using bankid 'on file', and request a mobile bankid. You'll get an activation code
9. Install the bankid app on your phone (e.g. bankid from the play store on android)
10. Start the bankid app on your phone, and type in your personnummer and the activation code. Create a pin, which is your password
11. You're done.
I mean, sure, it's a very awkward way of going about it, but it works and is conceptually simple.
Android in a VM
And there's absolutely no reason you can't use mobile bankid it in a virtualbox VM if you want to -- I used an android 4.4 x86 iso and created a virtual machine, and then set up bankid which works fine.