their loved ones like to say things like "heaven received another angel today". Where does this come from? Is it just a euphemism or do people actually believe this?

Some people really believe that. Others say it figuratively (perhaps the figurative use is what caused all the confusion in the first place).

For Christians, I'm aware of neither Biblical passage nor Church Tradition that teaches this.

Here is a nice, very short article by Billy Graham on the topic (including a couple of Bible verses that people may have interpreted this way)
http://billygraham.org/answer/do-we-bec ... to-heaven/

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In some people's minds, this might also be linked to Matthew 22:30
30 For in the resurrection they neither marry nor are given in marriage, but are like angels of God[a] in heaven

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