^^this. Never had an issue with engine start up using the ignition button. When remote starting the engine will turn over but not stay running. Do I need to hold down the fob button on the second press for it to receive the full signal? I usually double press and release the fob button. I’ll try holding down.
If your engine is turning over but not continuing to run on remote start, that is a problem...I thought you meant the vehicle was simply not acknowledging the command from the fob. If it is acknowledging the command by attempting to start, but not actually being able to do so...you have an issue that needs to be addressed by the dealer.
Related: at the very top of the fob there is a little LED light. When you "request" a remote start, it will flash green when the engine starts and red if there was some failure or if it was too far away for the vehicle to receive the command...I wonder what color your led flashes when you have this issue. (This feedback is also why the fob is so big.)
But in direct response to your question: no, you should not have to hold the remote start button down, a quick press should do it.