Since this is a public forum, we will hide your personal identifiable information, and I have sent that image of yours containing your personal identifiable information in a private message so we can work together to resolve the issue. Also, please check your private message in case you want to share another screenshot to avoid leaking your PII by click on the Private Message button right under your first post.
Based on your description, there is no clear indication that anything was done incorrectly. Email conversations can sometimes be displayed differently depending on how the email client groups messages, and seeing your own name associated with a conversation does not necessarily mean that the thread was changed or broken. Since the conversation is between you and Sidify Support, only Sidify can confirm exactly how they received your message and whether it was added to the existing support ticket or treated as a new email.
I recommend contacting Sidify Support directly and asking them to verify whether your latest message was received as part of the original conversation. They have visibility into their support system and can provide the most accurate explanation of how the message was processed.
As long as you received responses and the communication is continuing normally, the display name shown in the conversation is usually not a cause for concern.
In most cases, when you reply to an existing support email, your reply is added to the existing ticket or conversation as a response. Even if the conversation appears differently in your email client, it does not necessarily affect how the support provider receives or tracks your message. Therefore, there is generally no need to worry unless Sidify informs you that they did not receive your reply.
I hope above information is helpful to you.