I searched for an answer to this question, but I can't find an answer. I have a texture that I'm mapping onto multiple objects. I have it placed exactly where I want it on one face of one object, and now I want to copy those UV mapping coordinates for that face to a similar sized face on another object. The objects are otherwise the same shape. Any ideas?
You can use data transfert.
Let's take an exemple with a sphere and a icosphere.
The sphere is unwrapped. And we want to unwrap the icosphere similarly.
Place the icosphere at the same position. Select the icosphere and select also the sphere (in this order).
Now click on "data" from the "data transfer" section of the "tools panel". The choose "UVs".
You can also play with the severall options of the data transfer operator (if needed). Particularly, you can choose the UV map you want (the face you are talking about in your question) as source layer.