AFAIK, Im Eshkachech is a relatively recent addition. (As in, it only became popular since the founding of the State of Israel.) German communities have sung Tehillim 128 after the Kiddushin but before the breaking of the glass for centuries. We use the same tune as when we sing it Motzei Shabbos. It is either sung by 1, 3, or a group of individuals. (I have never heard of it being sung by the bride and groom themselves or by everyone around the Chuppah.)
As for why that specific Psalm - Google the translation. It is very apropos for a wedding ("Your wife will be as a fruitful vine", "may you see children [born] to your children").
Source: I am a Yekke. This was done at my wedding and I (quietly) do this at every wedding I attend. Also, Shorshei Minhagei Ashkenaz has a chapter on this.