It's common in conversational American English to say "I forget" when the intended meaning is "I have forgotten" or "I forgot".
Another way to look at this, which avoids relegating this use of "forget" to conversational English (a categorization which makes it sound as if this way of speaking were inferior, though tolerated in certain dialects), is to see that "forget" can be used with stative meaning or with dynamic meaning, just as "remember" can. Each use has a different meaning for both verbs.
(1a)
stative: I remember his name. It is John.
(1b)
dynamic: I have just remembered his name. It is John.
(2a)
stative: I forget his name. Do you know what it is?
(2b)
dynamic: I forget his name all the time. I'm going to have to write it down.
In their dynamic use, "remember" indicates the emergence of a piece of information in one's memory bank, and "forget" that a piece of information has left one's memory bank. In their stative use, "remember" indicates that a piece of information is present in one's memory bank, and "forget" that a piece of information is absent from it (with the implication that it was there before).