The multimeter in DC mode will average the voltage being measured if it is not DC. So, the 0.5V measurement could simply be due to the buttons being multiplexed, so the voltage across them is not constant but pulsed instead. You can check for this by setting the multimeter in frequency mode. If it does measure a frequency, then the buttons are multiplexed. You can also check how they're wired: if there are less traces going to the microcontroller than buttons, and if several traces go to several buttons, then they are multiplexed.
If it is multiplexed, you won't be able to just connect an arduino output to the buttons. This would be a bad idea anyway: if the arduino is powered and the machine is not, you don't want the arduino outputs to inject current into the machine's micro.
The 4066 analog switch solution will work in all cases (multiplexed or not) so it is the simplest. It also provides some form of isolation, not galvanic isolation but it will prevent the arduino from sending current into an unpowered machine.
74HC4066 is an easy to find 4-switch chip. Its internal resistance is about 50 ohms, that should work fine. Just wire one analog switch across each button. You need 3x about 50c chips for your 12 buttons, so it's not going to break the bank.
You must connect the sewing machine micro's ground with the arduino and analog switch ground. It's best to keep wiring short and tidy. Add a decoupling cap on the board with the 4066's.
Check the power supply for the sewing machine microcontroller. If it is lower than 5V, you're good to go. If it is higher than 5V (unlikely) then the analog switches should run from the higher voltage supply, with voltage translation. These switch chips can't handle voltage higher than their own power supply.
If you steal 5V power from the sewing machine to power your arduino, add a schottky diode to avoid powering the machine's microcontroller from the arduino's 5V, and make sure the power supply can deliver the required current.
Have you searched for an IO port on the machine? Serial perhaps... that could also be useful.