It is not common for a child not to sing along in school or to have a best friend, but it is not rare, either. As far as autism is concerned, I think she is way off the mark. Even a mild case of autism would have far more deviant behavior than you describe, and you would have been concerned much sooner.
The teacher is ignoring cues that would help her work with your son. For example, she notes that he doesn’t do things (cutting out shapes) by himself but that he can do them if she works with him. That should cue her to the fact that she needs to put him close to her when the children are working at a table, or that she should circulate around to him frequently if she is not seated with the children.
Teachers can also help shy children by occasionally assigning them to specific playgroups. “Steve, today you and Earl (a made-up name for your son) and Norma are running the store.” You might ask her whether she has ever done something like that. But you have to help, also: arrange for him to play with just one or two other children (at your house or theirs) and hang out close by in case he needs to be brought into some of the activities.
Finally, let me say that I don’t think I’ve done a very good job of answering your question because you haven’t told me a lot about other things that might be important. For example, does he talk well and understand others? How are his motor or muscle skills? Are there brothers and sisters in the family who are so competent that they put him down no matter what he does? Give me some more information, and let’s try again next month.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.