Is it polite to write your name next to someone else's business card?

This is definitely not good. If the company really doesn't want to spend the money, you can ask the company to make you a box of business cards at your own expense. If the company doesn't want to do it, you can do it outside by yourself, or ask the company to print some business cards with company address and telephone information. Only the business cards and template business cards that are not printed on the mobile phone can be used at that time, and other interns or probationers can use them. Your suggestion was adopted by the company, which also verified your ideas and abilities.