Hello,
It sounds like you are using a third party email address, such as Gmail or Yahoo, as the sign-in username/alias for your Microsoft account. Is that correct?
When your Microsoft account uses a non-Microsoft email address, it does not have its own email functionality. The actual sending and receiving of emails still happens through your third party email provider. As a result, when you send a meeting invitation through Teams, Microsoft cannot access your external email to send it. Instead, it sends the invitation using a randomly assigned Microsoft email address, one that ends in outlook.com. This is why recipients cannot reply to the invite directly. However, your meeting itself is not affected.
To give your Microsoft account proper email functionality, you can create a Microsoft-based email alias and set it as the primary alias. Here is how:
Go to https://account.live.com and sign in to your personal account.
Click on "Your Info," then choose "Sign-in preferences," and select "Add an alias."
Choose to create a new alias. It will allow you to create one that's outlook.com.
After creating it, set it as your primary alias.
Keep your current third party email as a secondary alias. Do not remove it.
You can then check emails at https://www.outlook.com. If you prefer, you can also enable email forwarding in the settings so that all messages go to your usual email address and you do not have to check multiple inboxes.