Is this sagging flat black wire over my garage roof a safety hazard?

That flat black wire is almost certainly a **fiber optic drop** or a legacy telephone line, not a high-voltage power line. Power service drops are ussually "triplex," wich looks like two insulated wires wrapped around a bare neutral wire. If it's a single, flat, stiff cable, it's definitely communication. These lines often sag over time beacuse the attachment hardware at the house or the pole slips, or the messenger wire inside the jacket stretches out after a heavy ice or wind event.

To figure out who owns it, follow the line to where it hits your house. It should terminate at a small plastic box, often gray or tan, labeled with a company name like AT&T, Verizon, or a local telecom provider. If it's going into a box that looks like it belongs to a phone company, that’s your target. If it goes into a larger metal box with a glass face and numbers spinning, that’s your electric meter, but it is extremely unlikely that a flat wire would be feeding your main panel. You’ll want to call the repair department for whichever utility owns that line and tell them you have a "low hanging drop" creating a hazard over a structure. Be prepared for them to take their time. Since it’s not a power outage, it’s usually low on their priority list. But if it’s actually rubbing against your shingles, make sure to mention that specifically. Constant friction from the wind will eventually wear through the jacket and the roof.

If you find out the line is totally dead—like if it's an old copper line and you're now on fiber—some people just snip them at the house and the pole, but I wouldn't do that. It’s technically utility property. Plus, even if it's low voltage, you don't want to accidentally clip something that's still active or mistake a power leg for a data line. Stick to calling the utility company and keep bugging them until someone shows up to tension it back up or remove it.

I just checked and the box on my house has an old 'Bell System' logo on it, wich feels like it's from forever ago. If the company technically doesn't exist anymore, who even takes responsibility for it?

I'm also worried because there's a big tree limb leaning near it. If that branch snaps and catches the wire, is it strong enough to rip the siding or the trim right off my garage? It looks like it's only held up by one tiny rusty bracket and I'd hate to have a hole in the wall over a dead wire.

Bell System usally means AT&T or whatever "Baby Bell" took over your local area. Those old drops have a steel "messenger" wire inside for support. If that tree limb hits it, the wire is often stronger than your garage's trim. Instead of the line snapping, it’ll likely yank the mounting bracket or the whole fascia board right off the structure.

I've seen it happen dozens of times—it mangles the siding and leaves a real mess. When you call, tell them the line is "mechanically strained" and threatening the building. They tend to move faster if they think they'll be liable for property damage.