How do I set different bicycle profiles on the 830? It was easy on the 800.

It does not recognize bikes.  You set up different activity profiles, each with maps, custom number of data pages, custom fields, etc.  You then select the desired activity profile for your ride.  Sensors are paired separately and the unit will use whatever senors are active when it boots up or when the sensors wake up.   I have two road profiles, two mtb profiles, and a trainer profile. All have different power meters and different speed/cadence sensing. I have several HRMs and it will us whichever is active.

You cannot move activity profiles between different models of Edges. 

I have four activity profiles for four different bikes, and I've named them as the bikes that I ride that for that profile.  So when I hop on a bike, I just select that bike's activity profile.  It essentially works the same as a bike profile.

I feel like they went backwards. Having separate bike profiles is so much better. I wish they would go back to that option.

Absolutely agree!

I so fully agree. At least Garmin should give the option.

I understand that some users prefer the previous setup of setting up multiple bikes rather than activity profiles. I apologize for any inconvenience this has caused. 

I can only suggest setting up different Activity Profiles for the different bikes you have. Since the 830 has a "pool" of sensors, whatever bike you are riding will pull up the correct sensor information when those sensors are engaged while riding. You will then be able to have different data field setups for each bike, as well as different autopause or autolap settings as well. 

This is exactly what I do.  It works the same as bike profiles.  I have an activity profile set up for each bike, and just rename the profile for whatever the bike is.  

The one thing I'm missing here is the configuration of sensors (particularly the speed sensor) per activity profile. If I want to move my speed sensor from one bike (or set of wheels) to another - which has a different profile - I also have to change the setting of the wheel's circumference for that sensor, the way things are now. If instead each profile would also include its own settings for this kind of sensor, you wouldn't have to do that with every relocation of the sensor.

The only way to avoid this hassle is to buy for every bike (or wheel set) its own speed sensor. I guess Garmin doesn't object to that ...

That is true.  But even if the Edge allowed the use of the same wheel speed sensor for different profiles, to me it would seem worth it just to have separate sensors to leave in place rather than transfering them between bikes. 

FWIW, you can use any Bluetooth or ANT+ wheel sensor; there are options other than the Garmin sensor.  In fact, on two of my bikes, I use older Polar Bluetooth speed sensors that I owned before I bought my Edge.  They work fine.