Can I use primers from different paint companies in the same room?

If it's latex primer, watch the smell - or just throw it out.

Especially now with the low-VOC regulations, mold or fungus can get growing in a can of water-based paint or primer. It creates a horrible smell (but one easily attributed to "the expected stink of paint".

You don't find out until after you've painted a wall that the smell isn't leaving. Ever. It will then defy every possible effort to seal it in, cover it up, etc. Now you have to remove it all. What a job! These nightmares have ended with people having to tear the drywall off the wall.

Primer's job is to create a uniform surface

So that the topcoat will behave the same all over the room - be an equally appropriate surface for paint, absorb the same, sheen the same, and have its color print through the same. Otherwise if you painted a checkerboard wall, the checkerboard would still be printing through after 2-3 topcoats.

So you want to arrange the primer job so that the final primer coat is essentially the same throughout the room. Whatever sequence of use makes that work best, that's what you do. If the characteristics of the three products are similar enough, that may be enough.

Mix products together? Not without factory consultation

Because even if they are the same type (alkyd vs latex) they may not be compatible chemically.