Is there a notification attached to this sound?
You say "siren" do you mean the phone sounds like a police car/burglar alarm/warning, or do you just mean it makes a notification sound?
You say it happens when you move and used the example of being in a vehicle, are there other times this happens? Is it at all speed or location related? For instance it doesn't happen when walking but does happen when running, doesn't happen when walking or running but happens when biking, doesn't happen when I'm in town but happens when I'm in the countryside?
It can be controlled with the phone volume, which volume control controls it? On Samsung you should have 4 controls typically, a wrench which controls system sounds, a bell which controls notification sounds, a speaker which controls call ringtones and a music note which controls media volume, when you change the volume which of those symbols appear?
Are you able to cause the issue to happen? Is there an event or task that you perform where this is sure to happen every time?
How long has this been happening? Can you think of or remember anything else happening around the time that this started? Installed a new app, scrolled through settings, opened developer options, dropped the device or got it wet... anything that comes to mind?
What app are you using for SMS?
What carrier do you have?
Is the device a carrier device or an unlocked device? Carrier devices usually have a splash screen with the name of the carrier, extra pre-installed apps, carrier apps that came pre-installed. Unlocked devices usually don't come preinstalled with carrier apps, often don't have any branding upon start-up?
Have you tried to reproduce it while the phone is in Safe Mode? If it happens in SM it likely is something related to the system, meaning we should be able to find a setting or something like that. If it doesn't happen in SM there is a good chance it is app related, which may have a setting still but it won't be a system setting. To get into SM press the power button then hold the power off button until Safe Mode appears and tap it to restart the phone in SM.
Initially because it happens when you move my first thought is something to do with SMS just because distracted driving is such a hot topic. I would imagine if it were related to something like that there would be some kind of notification that also pops up saying something about being safe and not getting distracted. I've never heard of an app that does this but it could be a setting or something accidentally turned on, especially when you think about how things are labeled. I could easily some weird setting name being turned on or tried and forgotten when nothing was initially observed. 10 seconds seems kind of long for that but you never know. My second thought was an accessibility setting. I'm picturing something for the visually impaired, not that they wouldn't be able to tell they were moving but all I'm hearing in my mind is the sound of a cross walk chime. Probably not but it's a thought. Similarly I can very easily see someone scrolling through their settings and turning on and off things to see what they do and maybe missing putting one back or leaving it enabled when it doesn't seem to do anything. I've also seen people with issues after allowing a child to play on or with a device. They can find anything and turn things on or off that we didn't even know existed.
Let us know what you find and the answers to our questions and we'll see if we can help you. Oh and before I forget, Welcome to Android Central.