It is good to keep in mind that in buddhist cosmology brahma-loka, or rupa-loka is a sphere of existence connected with jhanas, which are one highest states of consciousness in samsara together with arupa jhanas /arupa-loka.
Brahmas are very blissful beings, full of purity, loving-kindness, compassion, equanimity - and generally so called brahma-viharas (which means abiding of brahmas - divine abidings).
So in general, brahmas are embodiments of the everything the best in samsara, except for Nibbana itself.
So I think “Living together with Brahma” means they’re full or spiritual bliss, loving-kindness, goodness, wisdom etc. - all spiritual qualities.
Thinking of Brahma can mean something similar to “thinking of jhana” - so it generally means very blissful, positive, divine states of mind/existence, everything that is most good and divine.
My post is based a lot on what I’ve heard from Ajahn Punnadhammo - author of great book on buddhist cosmology and from Ajahn Brahmali - who has discussed topic of Brahma on many occasions, including on this forum.
Check out this video if you’re interested It might shed a lot of light on the topic of “Brahma” and why it is so connected to buddhist path. Long story short: jhanas are central part of buddhist path, and brahmas are beings who are reborn in brahma-loka because of attaining jhanas when they were humans. And to attain jhana, you must be very spiritually pure and advanced, and has a lot of great qualities that are required also for realisation of buddhist Awakening.
Also: in DN27 is a good track
DN27 - The Origin of the World
Then some of those same beings thought, ‘Oh, how wicked things have appeared among beings, in that stealing is found, and blaming and lying and the taking up of rods and banishment! Why don’t we set aside bad, unskillful things?’ So that’s what they did.
‘They set aside bad, unskillful things’ is the meaning of ‘brahmin’, the first term to be specifically invented for them.
Brahmin at the beginning meant all renunciates and people who renounced all evil. And before the Buddha, Brahma is sort of culmination of life of renunciate. And even anagamis are brahmas reborn in a brahma-loka called “Pure Abodes”.
So saying that someone “Lives with Brahma” means very pure and good life. And in fact Buddha oftentimes used such words, which meant he respected all people and beings who were connected to brahmic way of life. Buddhism is pretty much also renuncing of all evil and doing most good thing which is enlightenment. Buddha just got that one more step further and transcended even Brahma-Loka. But basically in most common sense as in DN 27, Buddha and his followers are also brahmins - because they do what is right, they live according to dhamma. They just took it to another level. Thats why I believe Buddha used word “Brahma” so respectfully. It means holy life and holy aspects of life in general.