Learn English – it called when someone acts one way then acts another way in front of someone else and gets you in trouble

A schtick refers to a comedian's style and choice of topics in general, and could include facial expressions or actions as well as words or style of delivery. It doesn't refer to a specific gag that is repeated often (although a repeated gag might be part of a comedian's schtick). It also often distinguishes a comedian's on-stage role from their actual personality:

Gracie Allen typically played a dingbat character, but that was just her schtick. In real life she was an exceptionally intelligent person.

A running gag is a joke that is repeated in a given work (book, film, tv show, etc.) or series of works. It's not necessarily spoken by the same actor at each occurrence.

"gag routine"(*) is not a commonly used idiom in English.

A signature gag would be immediately understood to have the meaning you suggest, but it isn't an idiomatic phrase, it's just two words used together that mean what you want. However, the phrase is "signature line" is probably more common than "signature gag". For example,

"Take my wife, ... please" is Henny Youngman's signature gag [or signature line].

A catchphrase is another related term, meaning a particular phrase used by a performer (not necessarily a comedian) throughout their career. This could include Henny Youngman's "Take my wife," or John Wayne's calling people "pilgrim". (Thanks @JoeSteele)

In the United States, one term is suitcase student, and an institution which has many such students may be known as a suitcase school. For example, consider this 2013 New York Times article about Central Connecticut State University. Connecticut is one of the smallest states, and CCSU is an institution sponsored by the state government, so the great majority of its students are from areas less than two hours away (which is extremely close by US standards), and so this pattern is prevalent.

Almost half of Central’s 7,700 full-time undergraduates live in dorms or near campus. But most vanish each Friday, joining the army of undergraduates at “suitcase schools” around the country who desert their campuses on weekends.

They head home for the same reasons suitcase students always have: favorite meals, moms (and now dads) still willing to do their laundry, high school friends and sweethearts, and jobs. The refrain “There’s nothing to do on campus” is self-fulfilling. …

^{Moore, Abigail Sullivan. "Off Off Off Campus" in The New York Times, Jan. 31, 2013}

The article is full of other suitcase terms, including suitcase culture, suitcase mentality, suitcase legacy. It refers to a student for whom life at the university residence halls is temporary, not a true home; therefore, they pack a suitcase of clothes, as if going on a vacation somewhere.

I wouldn't consider this term to be commonly used, as most colleges and universities are either traditional residential institutions or commuter schools, where only a very few students or none at all live independently on or near the university campus. Surprisingly, I found no results for it in COCA. In Google Books, however, they go back to at least the mid-20th century, e.g.

One problem that is bothersome today is the "suitcase student," who leaves campus on Friday afternoon and returns on Monday morning. ^{Treudley, Mary Bosworth. Prelude to the Future: The First Hundred Years of Hiram College. Association Press, 1950}

Ole Miss remained a pleasant headquarters but scarcely a community of scholars. It was, as the expression went, "a suitcase school." ^{Lord, Walter. The past that would not die. Harper & Row, 1965}