Both Bosch and Milwaukee use a size M coaxial connector. That's a 2.1mm pin and 5.5mm OD, is the most common size of barrel connector, and is found on CCTV cameras and all sorts of devices and wall warts.
The Bosch heated gear will run off any 12VDC source. I've powered my Bosch jacket off of a Milwaukee M18 power source, a car cigarette lighter plug, and a cheap SLA hobby battery. You can buy size M jacks with solder-on and screw terminals.
The Bosch jacket will not power up, however, unless you convince it that the Bosch battery holster is attached, which is easy to do. The holster outputs, and the jacket must see, two pulses to off after the jacket is plugged in. To run the jacket from a Milwaukee battery for example, you must plug in the jacket 3 times, leaving it plugged in the third time. You don't need to do it particularly fast.
The Bosch jacket uses about 850 mA of current on high. The current is the same on medium and low, but it pulses on and off, probably using a PWM.
The Bosch battery holster, once plugged in, outputs 12VDC to a female size M connector with no detection circuitry of any kind, except it does give the two pulses to open when first plugged in. It could be used to power up just about anything requiring 12VDC, within its current capacity.
The Milwaukee M18 power source outputs 12VDC to a female size M connector without any apparent detection circuitry. It could be used to power up just about anything requiring 12VDC, within its current capacity.