I am currently doing my master's research in this area by having individual's model emotional labels. For example, they are given a set of interchangeable pieces and told to make an emotion. There is some base understanding of how people interpret the meaning of forms, but directly labeling them as emotions is very limited.
Meaning and emotion can be derived similarly, but are not actually the same. They are steps in forming understanding or interpreting an object. You are correct that some colors have innate messages as marketing research has shown that green often results in an uncontrollable increase in hunger. Blue creates a sense of calm, but dark blues will often make people sad over time similar to red increasing aggression or aggressive behavior.
I believe these both can be verified with primate research as well. This interpretation is built off our lived experiences, yes, as many foods are green. However, an evolutionary argument is that like primates we are attracted to green and yellow as they indicative of food sources. Evolutionary bright colors on insects or reptiles often indicate poison and can increase alertness of the observer. These two ideas may be opposite one another, but bright colors always draw attention.
Physically, our eyes are also able to distinguish yellow and red from white and black backgrounds, but have a difficult time viewing blues and greens.
Taking all of these ideas into consideration, evidence of both learned and innate reactions are apparent. Psychology in general is always a combination of both learned and innate abilities and through extensive research we are able to distinguish which side upon which we rely most heavily.
To answer your question directly, the assumption is so far that the meaning of 3D forms are representative of a combination of both Gestalt and Semiotics as stated previously, but I believe personification may also play a role.
In order to interpret an object or form as carrying an emotion we must personify them to some degree. Therefore, the semiotic understanding of a round 2D form often results in an interpretive meaning of connection may instead imbue a meaning of happiness when personification takes place.
However, I am still in the beginning stages of my research and am unsure if I've missed an area of research that goes into these emotional labels further. I will update once I learn more...