As others have said, you're correct. Many native speakers have a habit of using "there's" when they mean "there are," probably because it's easier to say the one-syllable contraction "there's," and there is no contraction for "there are." Even highly educated people sometimes make this mistake in casual speech, though usually not if they're speaking in a more formal setting. As far as I've noticed, this only happens with the contraction. For example, you'll never hear anyone say "How many people is there?" That's obviously wrong. But in casual settings, you'll hear many native speakers say "There's three people."

Yes, you're right. Characters in TV shows often use incorrect English because many native English speakers do as well and the writers of these shows either want to mimic the way real people speak or just don't know the correct way themselves.

I agree with the other answers. It's definitely supposed to be "There are millions of people in this city." But, sometimes native speakers are lazy and slang influences us etc. Also, HIMYM is an amazing show. :)

You are correct, sometimes English speakers just use bad English :)

Its more similiar to "exist millions of people in this city." is a close context.