Your main problem seems to be that the player is basically playing the eidolon as an extra PC. Thus, in effect, you don't really have a six-member party — you have a seven-member party where one player is playing two of the characters.
Further, by hiding the nature of the eidolon and pretending that it's human, the player is, in effect, depriving the other players of any in-character reason not to treat the eidolon as an independent character and give it equal attention.
Now, since you and the other players seem to find this annoying and unfair, the obvious first step would be to have a talk with the player, explaining that they're hogging more than their share of the limelight, and politely asking them not to do it. Remind them that role-playing games are a shared activity, and that the point is for everyone to have fun and to get to participate equally.
The problem with this approach, of course, is that it depends on the player in question being sensible and/or polite enough to:
- admit that the way they're playing their character(s) might be causing problems, and
- give a damn about it.
You should still try this approach first, because you might get the player to just nod and say, "Oh, I hadn't thought of that. I'll let Eddie the Eidolon take more of a background role in the future." If so, problem (hopefully) solved.
However, let's assume that the player isn't so reasonable. They've designed their character in a way they believe to be within the rules, and pretty clearly they're having fun playing it just the way they intended.
In that case, one way or another, you're going to have to force the issue, or unilaterally find a compromise that everyone can live with.
One thing I would not generally recommend is adding house rules to limit the eidolon's abilities, at least after the game has already started. The player chose to play a summoner, and the eidolon is a pretty big part of that. Just telling them "sorry, I'm going to nerf your class so that you can't play it the way you intended" isn't going to make anyone happy at this stage.
That said, you still have several options to choose from, such as:
The in-game approach: Reveal the deception
Having the other characters realize the nature of the eidolon (or at least suspect that something strange is going on) might help solve the problem, or at least could give the other players the leverage they need to do something about it themselves, instead of just grumbling about it out of game.
There are several features of an eidolon that ought to make the other partymembers suspicious. Besides the matching forehead marks and the whole summoning ritual thing, don't the other characters never wonder about where the eidolon sleeps at night?
If the summoner is unconscious, asleep, or killed, his eidolon is immediately banished.
Also, while a humanoid eidolon may have its natural armor shaped to mimic a suit of armor, they can't actually remove that armor, as it's really a part of their body (and, of course, they cannot wear normal armor because it "interferes with the summoner's connection to the eidolon"). Assuming that the eidolon does (appear to) wear armor, wouldn't it eventually get a little bit suspicious that he never takes it off?
The out-of-game approach: Limit the spotlight
Just tell your player that he's free to play his character(s) any way he wants, but he's not going to get any more "stage time" than the other players get on average. Thus, any (real-world) time spent on the eidolon's actions will come out of the time available for the player's actual character.
The tricky part with this approach is that, to make it stick, you may need to start exerting a lot more control and authority than either you or your players are used to. In combat, you'll need to tell your players that they have a limited amount of time to decide their actions for each round, and that if they don't, their character will just stand there and do nothing. (There are tricks to making this work, like planning and even rolling your actions in advance. Teach your players to use them.) Out of combat, you'll need to start keeping track of how much stage time each player has had recently, and if necessary, get into the habit of simply interrupting stage-hogging players to ask the others "OK, what about you, Bob? What are you doing in the mean time?"
On the other hand, the up side of all this is that, once you and your players get used to a bit of time pressure, this approach can result in a much quicker and enjoyable game than spending minutes dawdling over every die roll, even if you don't have any actual problem players. It's just the transition that can be painful for everyone involved.
The rules-as-written approach: Division of control
While it says right in the class description that "a summoner and his eidolon function as individuals," and while the summoner can telepathically command the eidolon at any time and over any distance through its Link (Ex) ability, the rules don't actually say who decides what the eidolon does when not explicitly commanded, or how the eidolon decides to carry out any unspecified aspects of the summoner's commands.
Now, obviously, the usual way to handle that is to just let the same player play both the summoner and the eidolon, simply because that's usually the easiest and most comfortable way to do it. However, if the player is clearly abusing that privilege, there's nothing in the rules that says the DM can't assume control of the eidolon's actions, of course subject to any explicit orders the player gives.
What counts as "clear abuse", then? Well, that's up to you and your group to decide, but I'd say that treating the eidolon as a "second PC" certainly counts, or at least gets close. If the eidolon really is as active in the party as the player's actual character, then it should have an independent personality — if both the summoner and the eidolon are, say, having a conversation at the same time, there's no way the summoner should be able to control every word the eidolon says.
Mind you, this option might be most effective as just a threat — a simple offhand remark to your problem player about the fact that the rules don't say who gets to directly control the eidolon just might be enough to get them to reconsider their playstyle.
On the other hand, it could also be that the player might like having the DM play the eidolon like that; it certainly could produce some interesting dynamics, if the summoner and their eidolon may occasionally have different ideas about what "take out that orc" or "convince the guards to let us pass" really means. Who knows, it might be worth trying.
Ps. See also PRD: Companions (Ultimate Campaign)
The last-resort approach: Drawing the line
Finally, if nothing else seems to help, remember that it's your game, and you don't have to put up with any behavior you consider harmful to the gaming experience. If the player insists on hogging an excessive amount of attention, refuses to yield the stage voluntarily and complains (or causes trouble) when you call up other players instead of him, you have the right to tell him to play nicely or stop playing.
Of course, hopefully things will not get that bad. Still, if it looks like they might, even just sitting down with the player(s) and saying that you've considered this option might serve as a wake-up call. Or it might not, but at that point, what is there left to lose?
