Because the celestial eagle summoned by the celestial commander understands him,

Yes, the Eagle Can Aid Another in Combat

According to the spell summon monster I et. al. the summoned creature "attacks your opponents to the best of its ability. If you can communicate with the creature, you can direct it not to attack, to attack particular enemies, or to perform other actions."

The extraordinary ability divine tongue allows the celestial commander to communicate with the summoned creature, no matter its Intelligence score. Therefore the summoned creature can, according to the spell summon monster I, be commanded to aid another in combat without the celestial commander needing to resort to a Handle Animal skill check (DC 25 or DC 27) to push the animal to perform a trick it doesn't know (e.g. the trick aid), which is what other casters must do when they want summoned, unintelligent animals to do their bidding when their bidding isn't biting baddies.

The Secrets of Adventuring
If you're going to use that archetype, and you haven't already, I encourage you to purchase Rite Publishing's Secrets of Adventuring, not because I have a relationship with the company (because I don't have any relationship with Rite Publishing), but because the text might provide further information about the archetype--and maybe even that ability in particular--that's not contained on d20PFSRD.

While a summoned monster does initially act on the turn of the person summoning, it has its own initiative track. If later it delays its action, or uses a spell or ability that changes its initiative, its initiative changes to a different count than the summoner's (or vice versa).

The rule you're quoting only says that it gets to act on the turn it was summoned, on the same initiative as the summoner, not that their initiative can't change individually later on.