This product is using the exact same ID numbers as Teensy, which means it either has a Teensy inside, or similar hardware and is running code made with Teensyduino (which is not unusual - lots of people do use the code to make their own products), or someone went to a lot of trouble to precisely mimic the ID of a Teensy on completely unrelated hardware (which seems very unlikely - that's extra work for no real benefit).
Now for some guesswork....
This product probably was designed with Teensy and then made with the same chip and uses all the same USB code, but they probably didn't put the bootloader chip on their board. I'm also going to guess they didn't disable the _Teensyduino_reboot_() function in their code. If this is true (remember, I'm guessing) then their product is probably trying to go into bootloader mode when Arduino finds it instead of your Teensy. If this is true, then you MUST NOT CLICK UPLOAD. You will need to click Verify in Arduino, and then physically press the button on your Teensy. Also make sure you close the Arduino Serial Monitor before you press the button!
To make matters even more confusing, those screenshots look like Windows 7. I don't see anywhere in this thread which version of Windows you are using, so this is more guesswork...
All pre-10 versions of Windows have a terribly confusing USB serial driver bug. Normally Arduino avoids it because the serial monitor automatically closes. But in this case, if another program is using that board's COM port and Arduino mistakenly causes it to try to enter bootloader mode, then you can his this issue:
This is really confusing, because the problem manifests later (even hours or days later) when you plug and reconnect the device. If you have 2 serial devices and the wrong one is rebooting (or getting stuck trying to reboot), and you have Windows 7, then you are going to hit this driver bug over and over. It's very confusing in practice.
On Windows 7, all you can do is very VERY CAREFUL to always first close the serial monitor. Then click Verify, not Upload, and press the button on Teensy after the code verifies.
If you are able to open that pinball maching controller, I would be very curious to see a photo of the circuit board. If you contact the company, please suggest to them to use their own VID/PID numbers (or just make some up that aren't the same as Teensyduino), and if they did use the Teensyduino code without the bootloader chip, send them this advice to at the very least comment out the _Teensyduino_reboot_() function in a future version of their firmware, so their product doesn't lock up when someone with Arduino+Teensyduino tries to upload to a real Teensy.