I would like to send a video massage from my gallery to someone on Tango, but when I press share I am not getting the icon for Tango chat, like I do when I want to send a photo from my Gallery.
By default you should have the possibility to send a video as a message on Tango. If that's not possible, then it means you don't meet something on the support page as specified by the developers.
I recommend you visit the following page to learn about the conditions you have to meet before sending video messages, and then you can try again.