1). Institution:

Coming from the Latin words In and Statuo.

In – deriving from Proto-Indo-European *h₁én .

*H₁én – To be indicating location, inclusion, or position in relation to something spatial or temporal quality or action.

Statuo – Derriving from  stō, from  Proto-Indo-European *steh₂- .

*Steh₂ – To stand, action of standing. To stay, remain, action of staying, remaining.

So in conclusion the word institution could be meant to be indicating relation to the action of staying standing.

2). Institution is the name we have for organizations (even if the organization is only one person)  that becomes so established that the name of the institution itself becomes a separate identity from the people working or organizing it.