The popup shows if I activate the button, it only shows on Firefox not Chrome. I use several forums that are Tapatalk enabled, they give a popup that shows once as you described, doesn't show again unless cookies are cleared, very common behaviour.
However this is the only forum that I visit that shows that button on my browser, without my permission and with no apparent way to remove it.
Quite concerning, especially given Tapatalk's reputation for somewhat shady exploits, possibly why Google have removed Tapatalk from the Play store.