You can donate to a temple? But for what cause? Does it do anything?

I don't think it does. And imo it doesn't have to do anything for the player either. But it would have been nice that if you give enough donations, that the temple than adds another statue and more generic 'priests of god X' show up as well as 'worshippers'. For those that roleplay a character with loyality to a certain god it would be a small yet nice addition.

Apart from feeling warm and fuzzy inside, AFAIK nothing.

You do get a brief boost in Speechcraft when donating to a beggar, do you get that when you donate to a Temple? I think you should at least get the Blessing for the temple donation. Then again, you can role play that, in that you have to leave an offering before you take the blessing.

Just like a real life church, nothing

Usually such donations are given for the church to be able to pay for maintenance and priests. In a land where the church is not under the governmern (like we in Norway got) the churches got to take care of themself.
I guess it's like that in Skyrim too.

I like this guy