It would make sense for trains that use both lines to have both logos on the side, although that might also present some issues and causes for confusion I suppose.
The way FCC did things seemed to work fine. One company, one brand, but two distinct routes within that. Govia has another approach and it will be interesting to not only see how they do things on day one, but how it evolves (and if they make changes and tweaks that they didn't intend to based on feedback).
I am not sure that with the two routes being connected properly in 2018, yet still being very much separate for a LOT of people on their daily commute, it works to clearly mark them as one service or another. Govia must hope that the use of the same logo design, but with different colours and text, will be seen as obviously connected by everyone.
With two websites and two Twitter feeds, it may have many people beginning to think the lines have totally split from when FCC was running things, and are now separate franchises. I assume the two websites will be one and the same, but with a different home screen based on the URL used.
But Southeastern has different sub-brands for services, and back in the day we had differently marked rolling stock for Thameslink, so I am sure it will all work out.
And where it doesn't, they'll be called out on it here for sure!