Is it incorrect to shorten "my hobby is…" to "my hobby's…" or does it just look strange?

Generally, when writing, you should avoid all contractions. Over using contractions can make the writing harder to read and meanings my bedroom ambiguous. The reader can contact themselves if reading aloud and some would rather not contact if such is their normal style.

An exception is when you are trying to write to imitate speech such as within a quotation. In this case you would contact the writing to match the speech even if the contraction is unusual.

When speaking, native English speakers would contact “is" in most cases without even knowing they are doing it.