The rules of eating a meal in the same place one made or heard kidush are written among the rules of Friday night's kidush (and applied to both). Thus, the rule (Mishna B'rura 273:25) that cake suffices for this (so one need not immediately eat bread) applies to the nighttime as well as the daytime kidush. (However, even if he is famished during the day and cannot eat cake or the like so eats fruit, he can't do so at night: cake or the like is required at night. MB 273:26.)
Why, then, don't people do this in synagogues? I suspect it's practical considerations: women aren't there (in many synagogues, and, in any event, traditionally haven't been there, and old habits die hard), so it's unusual to make a celebration in the synagogue, for the community, Friday night. Moreover, people with children want to go home before the children need to go to sleep. And similar considerations.
Moreover, there are two specific reasons to have a kidush with no bread in the morning that don't apply at night. One is that (in many synagogues) the prayers end before lunchtime in the morning, and people want to eat something light. The second is a specific custom (Taame Haminhagim, asterisked footnote to 299) to eat "four meals" (the second of which requires no bread) on Shabas, so that some people deliberately (even at home) eat something light and take a break before lunch even if it is already lunchtime.