“There are three of us” is correct and much more common in English when speaking about a number of people in a group. “We are three” is often the result of someone translating from another language or an older form of English.
When we want to say something “exists” in English, we use “There is/There are”. In this case, we could be talking about the number of members in a family or any group of people.
There are 5 people in my family.
There are 11 people on a soccer team.
There are(+number of people in a group)
When you want to say the number of people that are present or included in a group, you should use “there are”.”There is” is not common in this situation because “there is” is singular.
There are 25 students in my class.
How many people are coming to dinner? There are four of us.
There are 6 people living in the house.
If you are talking about a past event, you should use “there were”
There were 15 people at the party.
There wasn’t anyone there.
We are + number
“We are+ number” is possible in English but can sound a bit strange.
The main situation it might be accepted is when you are describing the number of people that are going to eat at a restaurant. The contraction(we’re) is always used and this is very informal. I would not use this but you might hear it in American English.
How many of you will be having dinner? We’re four.
I would usually say “There are four of us” and I think if you have questions about this, it is better to stick to “There are”
In other situations, “we are+ number” sounds very strange and awkward like in the other examples above.
The reason for the use of “We are 5” is because in many of the other languages around the world(French, Spanish, Indian languages) that would be the direct translation.
English is an ever-evolving language and was also derived from French which was an official language in England for about 300 years.