It’s possible, but needs to be engineered for safety, and that design/testing/certification will increase the cost and complexity.

You can have solar panels and a battery totally off grid, where the big battery just acts as a generator, with its own inverter creating AC power for anything you plug in. That’s really simple and cheap, but isn’t safe for connecting to and powering a grid-connected house circuit. So anything you want to power with one of these systems needs to be plugged into outlets that only get their power from these batteries.

You can add a grid-following inverter that safely matches the grid frequency AC, so that you can use the solar power you collect in your own normal home circuit, to power your own household appliances. But the simplest design here is a grid following inverter that doesn’t work when the grid isn’t connected. It can only add to something that already exists and can’t do things on its own.

If you want to do both, where it can work without grid power and it follows the grid when the grid power is on, you’ll have to design a system that can switch between the two modes without delivering power where it’s not expected or generating power that conflicts with the grid’s AC waveform. Making it automated, like an UPS system, is even more complicated.

It’s not impossible, or even that difficult, it just does add complexity and the engineering tradeoff is always the question of “what problem does this solve, and is solving that problem important enough to devote these resources to it?” For anyone on a reliable electric grid where power outages are rare, the answer is usually no.