What is this anime about people who can transform themselves into giant robots with the help of a device/talisman?
The series that OP was looking for is Moldiver, a 6-episode 1993 OVA anime series.



The female protagonist is Mirai, the brown-haired girl in the middle. And the glasses guy on the lower right corner is Hiroshi, her brother and inventor of the Mol Unit - the transformation device.

Plot summary by ANN:


  Hiroshi Ozora invents the Mol Unit, a device that makes its user invincible. His plans of becoming a super hero are short lived however when his sister, Mirai, accidently modifies the suit configuration. Now she is forced to save Tokyo by herself from the evil Machinegal and his gang of all-female androids.


Below is the screenshot 18 minutes into episode 2 showing Mirai getting her clothes ripped off when transforming with her clothes still on.



Below is a series of screenshot of Hiroshi getting naked inside the tunnel of the stadium to prepare for the transformation but failed (17 minutes into episode 2), after which he contacted Mirai and asked her to transform (above). When he was instructing Mirai (21 minutes into episode 2), a dog came and took his clothes away behind him.



There wasn't any scene of him chasing the dog, though.



When I first tried to identify the anime in question, I checked through the list of anime and manga in Clothing Damage trope, and Moldiver's screenshot kind of matches the description for the female protagonist.