Yes, with qualifications, that metaphor is easily understood.
- It is better to use natural English grammar and to say 'A taxi is a business card for a city'. English requires an article for singular nouns. ('of' wasn't wrong, 'for' just felt better to me. But then the more natural way to say it would be to use plurals anyway:
Taxis are the business cards of a city.
- There might be more salient terms than 'business card'. The Chinese use of 'business card' is already metaphorical. I think a term you mentioned really captures it:
Taxis give the first impression for a city.
This isn't particularly metaphorical, but gets the idea across really well and could assuredly work for other items "An airport gives a city's first impression." "A senator gives the first impression of a state"
These aren't common phrases or existing patterns (as far as I know) and I can't think of an similar kind of phrase that people use commonly, but it certainly is a very natural and understandable idea.