What happens when a commander is exiled by a player other than its owner — can the owner move it to the command zone?

In the Commander format, a special rule applies to commanders that would move to any zone other than the battlefield: the owner may redirect that move to the command zone. This applies to the library, hand, graveyard, and — critically — exile. It does not matter who controls or owns the effect causing the exile (CR 903.9a).

So if an opponent casts Swords to Plowshares targeting your commander, you as the owner may choose to send your commander to the command zone rather than exile. The exile event is replaced entirely — the commander never actually enters the exile zone (CR 614.1, replacement effect). You still gain the life from Swords to Plowshares, because the commander was targeted and the spell resolved; only the destination changes.

Each time a commander is cast from the command zone after having been redirected there, its commander tax increases by {2} per previous cast from the command zone (CR 903.8). The redirection is a choice, not mandatory — you could let it go to exile if you wanted (for example, to avoid increasing the tax or to satisfy a condition).

Concrete example: Your opponent plays Teferi's Protection, phasing everything out, and then later casts Bojuka Bog targeting you. Your commander would be exiled from the graveyard — you may instead place it in the command zone. It was never in exile, and next turn you may pay its mana cost plus any accumulated commander tax to recast it.

This rule is a Commander-format-specific rule found in CR 903.9a and referenced in the Commander rules document, and it supersedes normal exile mechanics for commanders specifically.