Sonic and Blaze's relationship is a whole lot of compromising. Sonic has his dimension, and Blaze has hers still. They still have responsibilities to their worlds despite their loyalty to each other.
The barrier that used to separate them isn't a problem anymore like it was in the old days, so their partings aren't forever. It depends on how chaotic or peaceful their respective dimension is and that factors in just how long Sonaze stay together.
Its a yes and no sorta answer that they live together, its complicated, lol. Sonic is still Sonic as well, so he has wanderlust and a need to travel, but he always comes back. Blaze's dimension tends to be treated as home for Sonaze due to how Blaze as empress can't really afford to leave her empire unattended.