I thought sammen meant "the same"
No, that's det samme
happily it doesn't mean semen.
Why does the German word for together, "zusammen", get the extra "zu" added on?
There's no answer to your question. It's just the language. But in general, in German "zu" is a very frequent preposition (like English "to") and many words start with "zu-".
In Swedish it is "tillsammans", so Danish is the exception here. :-)
How would "at the same time" be?
"at the same time" would be "på samme tid".
Vi spiser og drikker på samme tid. - We eat and drink at the same time.
There is a difference between "at the same time" and "together".