Well, I of course can't be 100% positive, but a horn speaker mounted on the firewall is typically how the siren for an aftermarket alarm is installed.
This is a typical car security alarm kit:
I don't know anything about how your alarm works, since we don't know who made it and what model it is. Maybe you could look under your dash? I'm sure the alarm installers tapped some wires at the ignition switch.
You're right that it might be going off for a reason. If it is indeed a security system, then it's going off because it thinks you're stealing the car. Any chance you were given a remote key fob or anything with the car that you could use to disarm it?
In theory, it's possible that simply disconnecting the horn would make the system prevent your car from starting or running. Some of the more clever systems do that kind of stuff if you cut the ignition switch wires when installing them. To properly remove the alarm system you'd have to reconnect your ignition switch wires to how they were (if they were indeed cut).
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