This is perfectly correct. You notice the present state / situation, even the the though the umbrella’s disappearance took place in the past. Notice the use of the present perfect “he has taken your umbrella” — this establishes a clear connection between the past action of taking the umbrella, and the current state of the umbrella being somewhere else.

It's acceptable usage to say : "I see that + clause" The "seeing" is in the present. What you are seeing is not stated but is evidence of a situation which takes place in the past, present or future (using any appropriate tense). e.g. I see (from the newspaper I am reading now) that West Ham United won its last match. e.g. I see (from the clouds outside now) that rain is coming. e.g. I see (from what is on the table now) that you have bought fish and chips again. You can replace "see" with "notice". However, in my experience, "notice" is more commonly used in the present perfect and past simple because it is a short action which usually you only report after the event. This seems to match your understanding, I think.

"I notice that he has taken your umbrella by mistake." - this sentence implies that the speaker notices in the present (even though the action happened in past) If the speaker had taken note in the past (at the time of the incident) the sentence could read. I noticed that he took your umbrella by mistake. (if the speaker noticed at the time of the incident) You could use the past perfect if the speaker noticed after the incident took place. "I noticed that he had taken your umbrella by mistake".