Very little is known of Shakespeare's biography and even less of his inner psychological character. We do know that he was married to Anne Hathaway and had three children with her. In terms of the nature of the marriage, we know only facts such as dates, the names of the parents, very limited financial information, and the names and birthdates of the children. No gossip is preserved, and even had it been, gossip is not reliable.
Shakespeare's plays usually took pre-existing plots and embroidered them, and thus the plot details of the plays reveal little or nothing about Shakespeare's personal feelings or temperament.
Although the sonnets are written in the first person, again, the form is a conventional one and we have no certain external evidence that would enable us to determine whether the narrative voice of the poems was simply a conventional narrator or represented the feelings of Shakespeare himself.
There is no element of the corpus as a whole that would indicate an radically abnormal or unconventional personality for his period; thus the best that we can conclude is that we have no evidence that he was unusually insecure or jealous.