For the Karoo it's a "yes", as long as you remember to mark which bike you used after you end the recording.
That looks good.
Also thanks for the pictures.
It's all manual, of course, which is a bit of a drawback.
Question is whether what you’re looking for actually exist (without the focus on workout and navigation features)
I've just looked around, and as I expected Sigma are still making exactly what I want - at least their basic devices support two different bikes (the descriptions of the fancier ones are more vague). From the current lineup, the BC 16.16 looks quite good. Or perhaps the Rox 12.0 Sport from the GPS'd line - but that seems to be not available anymore.
My experience so far is quite decent - I've had a few iterations of Sigma Sport BC xxxx since the late 1990s. Currently a BC1609 (I think) from about 10 to 15 years ago that has a cabled speed sensor on both my bikes - and does total mileage and total ride both for each of them,
and the sums.
I just thought I could go for something fancier, with GPS navigation being the main feature upgrade. This would be great for rides in unknown territory. Of course I can run Komoot or any other navigation app on the phone, but I do like a dedicated device on the bike (and a real speed sensor that's not GPS). Also, audio streaming to bluetooth earphones, GPS navigation/tracking and screen on drain a phone battery quite fast. And I don't feel comfortable having my phone constantly exposed to the bike vibrations on a handlebar mount - when it's in my pocket, my body acts as a suspension, so to speak.
For me it's the question of either sticking with a "simple" and cheap bike computer for the basics, and missing the fancy stuff, or going for a €400 all-inclusive solution which would be an awesome toy to have. But for that money, it definitely must have the basic features of a bike computer. And tracking the total mileage of the vehicle is really basic. Would you buy a car without an odometer? That's the level of implicitness on which I'm thinking about this.