How Do You Decide Between Analog Chip Manufacturers?
There is no single rule that drives you to just one analog chip manufacturer's parts. (Not in microcontrollers, either, despite your claim that only Atmel makes sense for you. But that's a separate question.)
A really good component engineer could probably fill books about how to choose a chip and/or manufacturer. I am not a component engineer, but I can lay out a bunch of cases where I've been forced to one manufacturer or another:

Unique chip: TI's TLE2426 is truly unique. There are numerous ways to build something similar out of generic components, but if you need that function and it has to fit in a TO-92 footprint, you're stuck with TI's chip.

No standard pinout: Unlike with op-amps, analog buffers never really settled on a common pinout, and there are few enough of them available that it's rare for more than a few to be really suitable for a given circuit. Say you pick NatSemi's LMH6321 for some reason. Even though another chip may be an acceptable alternative, no other analog buffer on the market shares that particular pinout, so as soon as you make a PCB with the LMH6321 in mind, you cannot change to another chip without re-spinning the board.

Subjective matters: In high-end audio circuits, a good ear can distinguish the sound of op-amps, or at least, op-amp families. If you happen to like the Burr-Brown sound, you will be looking only at their chips, while another will swear by Analog Devices chips, and you may both wish to ignore the Linear fanbois.

Price: Maybe you need an LM324 but aren't too picky about quality. In that case, you're going to be looking at one of the second-source suppliers for this chip like Fairchild or ST, rather than the chip's creator, NatSemi.

Durability: This is the inverse of the previous point. I'll bet anyone in the industry for more than a few years could tell you stories where the design engineer chose something widely second-sourced (an LM317 for example), did all their prototyping with a high-quality implementation of that chip design, sent it off to the manufacturing folk who substituted a far cheaper generic, and then had to deal with field failures because the replacement, though claiming equivalent performance in its datasheet, turned out less durable when abused in ways you cannot control from your clean lab. (ESD in winter in North Dakota, idiot end users unplugging connectors the manual says not to unplug while power is applied, etc.) It is often cheaper to go back to using the more expensive first-source version than re-spin the board with protection components scattered around the cheap knockoff.

Obsolescence: You're tasked with stuffing boards you can't re-spin for a design made 20 years ago, and some of the chips aren't available from their original manufacturers any more, or at least aren't available in the packages the design engineer chose 20 years ago. You might be forced into the arms of a company like NTE or Central Semiconductor, who specialize in offering old designs that all the tier 1 companies have abandoned.

Higher performance: A highly popular IC like the LM317 often inspires pin-compatible follow-ons with higher performance. Linear Technology specializes in this. If you find yourself needing better performance, it is often worth paying the higher price for a less common pin-compatible replacement than redesigning your circuit to give better performance with the original chip you selected.

Availability: Once you've run the gauntlet above, you might still seem to have a few choices of sources for a given part, but one of the sources has burned you in the past on availability. That sort of thing can force you to another manufacturer all by itself.
A more extreme variant of this situation is that you've whittled away your choices until you end up looking at just one part, but it's made by a manufacturer that doesn't always have ready stock. That might force you to abandon that choice, desirable though it may be on paper, simply to switch to a more reliable source. You might even end up reinventing that perfectly good wheel with generics just to get better control of parts availability.