Yeah...lots of questions about it...
1) Does it have a name?
2) Why was it invented, or do the Andalites not have a natural ability to morph?
3) Does every Andalite have one?
(need to re-read books, badly)
The others have already covered these questions pretty well, but what the hell:
1) Escafil Device. Presumably named after the creator of the technology. In Hork-Bajir Chronicles (when the technology was still very new) Aldrea says she had a friend whose mother was one of the designers of the device.
Ironically, Aldrea also said that at the time, many Andalites still doubted the usefulness of the technology.
Even later on in the series, we know that the Andalites used the morphing power for spying and little else. Considering the vast potential of the power, it seems rather foolish of them not to use it more, even if a lot of their battles were fought in space ships.
2) Supposedly created for spying, but not used at all by most of the Andalites met in Andalite Chronicles, Hork-Bajir Chronicles or #18. And no Andalites cannot morph naturally.
3) Aside from Estreen dancers (which are only mentioned in passing) the morphing technology was strictly for military personnel. Ax doesn't mention it when discussing his school days because he didn't have the power at the time. We can assume due to the war, Andalites were allowed to start military training at a young age. We know they are given basic morph training when they become cadets, but it is unlikely that Ax had used the power afterwards (until #4 of course).