Yes - they are very close synonyms. Sometimes they are not all natural in certain contexts. e.g. you normally "repair" or "fix" a car or a computer, but people don't often say "mend" a car or computer. "Mend" is used physically e,g. a garment, but also figuratively e.g. mend a broken heart, mend fences (fix a broken relationship).

"Fix" is a good all-purpose word. You can use it everywhere. "Repair" tends to be used with machinery (cars, computers) and buildings (houses, barns, fences). "Repair" goes with "broken"--you repair something that is broken. I'm a U.S. speaker. "Mend" is used a little more commonly in the U.K. In the U.S. "mend" is used with things like paper or fabric. "Mend" goes with "torn"--you mend something that is torn. You can also "mend" relationships with a person... it is short for an idiom, "to mend fences." It comes from an old custom that mending fences between farms is a joint, cooperative responsibility.