Well, that's true.
Doesn't matter how flat your organization is or how much friends you are with your teammates - as the manager, people are looking at you.
They look to see how you handle good news, bad news, challenges and confrontations. They look at your communications with other, within the team and outside of it. They look at you arrive, leave and procrastinate. They look at everything. Kinda like your kids do, but with more judgement..
Now that's not necessarily a bad thing. Not at all. You just need to keep that in mind and embrace it.
I'll try to give some examples for situations you may want to give extra attention to this matter:
During a rough patch:
When your team has hit a rough patch - whether it's a team member departure, overwhelming production issues, whatever - this is your time to shine. Even at a low point, knowing that people are relying on you - will help you keep your head held high. As the manager, set the example. Keep the good spirit, be focused, pick up the slack and in general - be a star especially at this time. Your team is looking and it will easier for them to follow once the example is set.
When getting praise:
Occasionally, you or your team will receive extraordinary praise. Maybe an executive will mention you in an all-company meeting, maybe you'll be on the news and people all around you will pop in to congratulate you.. Here again is a big test for you as a manager. Praise your team. In public. Don't let the success get to you only, make sure everyone share a part of the glory. Be modest. Enjoy your moment but know when to regroup and go back to normal. Keeping in mind that your team is watching you will have you achieve that and your team will appreciate your maturity.
When you're pissed off at someone else in your company:
Someone pissed you off. You're back from a meeting which went really really bad and want to blow off some steam. Well, Don't. Not in front of your team. No matter who is the other person, bad mouthing them to your team is wrong. It will get them a feeling that it's ok to do that, and they may be a bit less tactful doing so. It will also shatter the concept of "One Team" which is something you really want to keep in your team. Lastly, it will demonstrate to your team a wrong way to handle difficult situations or difficult people - getting pissed off means that you don't have an ability to solve the issue.
In conclusion, like in the image of the angel and devil on each shoulder - keeping in mind that your team is watching you and you're a role model no matter what you do, will help you keep your behaviour in check and even take advantage of situations to create "teachable moments".