I left my umbrella in shul. About a week later I found an umbrella that looks like mine in the coat room where I had left my umbrella.
Q: May I assume that the umbrella I found is mine and take it?
A: Many times it happens that someone loses something and then finds an item that looks similar to the one he lost but cannot honestly state that it is his own. Two discussions in the Gemara have a bearing on this question.
The Gemara (Pesachim 10a) discusses a mouse with bread in its mouth that runs into a room that had already been checked for chametz. The owner follows the mouse and discovers some bread. May he assume that the bread he found is the same bread that the mouse brought into the room, or is he required to search the entire room again for chametz? Poskim debate whether one may assume that the found bread is the same bread the mouse brought in. Some contend that if the owner already nullified his chametz the potential violation would be Rabbinic, and one may assume that he found the mouse’s bread. 
Others contend that even if one did not nullify his chametz so that the potential prohibition is Biblical, one may assume that he found the mouse’s bread (O.C. 439:2; Noda BiYehudah 1, E.H. 46).
On the other hand, the Gemara (Gittin 27a) addresses a delivery agent who lost a get. The Gemara rules that if he finds it immediately, the get remains valid without concern that the get he found belongs to another couple. If he did not find the get immediately and cannot be certain that the found get is the one he lost, he may not deliver it to the intended woman, because we are concerned with the possibility that it is not the get that he lost. Clearly, this Gemara indicates that for Biblical matters we are concerned that one did not find the object that the agent lost (E.H. 132:4). Poskim discuss the difference between the two gemaros; those distinctions have bearing on your case.
Some authorities note that these discussions are not applicable to your case. Although an object’s location is a siman (identifying feature), nevertheless, when someone claims to be the owner by identifying its location, we rule that knowing the location is not a siman since others also place similar objects there (C.M. 262:9). Therefore, stating that your umbrella was lost in a shul coatroom does not prove that you are its owner (Minchas Yitzchak 3:17).
There is, however, a rationale that permits you to take the umbrella. The umbrella does not have a siman. Consequently, if the owner already realized that his was lost you may certainly take it, since he despaired (yei’ush) of recovering it. Moreover, even if it is possible that the owner does not yet realize its loss (yei’ush shelo midaas) — and generally in such a circumstance one may not take the lost object — in this instance it is permitted. That restriction is limited to objects that one knows is not his. Therefore, if the owner proves the object is his, it must be returned.
In your case, since it is possible that the umbrella is yours, and since it does not have a siman, it will be impossible for anyone else to prove that he is the true owner, so no one would be able to take it from you. Furthermore, there is no concern for a hidden siman because if that was a concern, one would never be able to keep a lost object. Thus it is permitted for you to take the umbrella (Pischei Choshen, Aveidah 3:18).