Yes, it's safe to take more than 1 vaccine in the same day. For travellers who need vaccinations, it's common for doctors to administer multiple shots (regardless of your age).
Evidence has shown that getting several vaccines at the same time does not cause any health problems.
Your body's immune system is continuously challenged by external stimuli called antigens, which can be either breathed in or "eaten". It responds by making antibodies and disease-fighting immune cells. The specific antigens given in vaccines make up only a small portion of the daily stimuli that your immune system has to deal with.
Vaccines thus work by promoting the production of protective antibodies or immune cells.
Multiple vaccine shots will not overwhelm your immune system, and will not cause you to become more prone to falling sick.