Gundham is never seen with a glue stick in the entirety of the series, nor does he ever once mention glue. Glue, in canon, is entirely irrelevant to Gundham Tanaka.
Gluedam came about during an instagram livestream off thatguyinthebowlerhat's page, in which he and Gundham's English voice actor, Chris Tergliafera, cosplayed as Gundham and discussed Gundham-related trivia. During the livestream, Morgan Marquette (thatguyinthebowlerhat) was shown applying glue to Chris' eyebrows in order to achieve the eyebrowless look Gundham canonically displays.
Because of this, many of the viewers of began making puns referencing 'glue' and 'Gundham' together, until it devolved into a ship. The fans of the new pairing seemed to agree that glue and Gundham were destined to 'stick together,' and began to draft fanart of the ship.
While primarily being present on Instagram, Gluedam has managed to find its way onto other mediums, such as Tik Tok and Tumblr.
Some went so far as to extend the ship to include Sans from Undertale in order to rival his other crackship with a Danganronpa character. These fans claimed 'Gluesansdam' was far more powerful than its rival. This has yet to be seen.
In Morgan's February 13th livestream alongside Kane (changelingprincex,) the two of them publically explored the potential cargo dynamic a second time by doing a livestream tarot card reading of the chemistry between Gundham and Elmer's glue. During this reading, they uncovered that the two would have 'a passionate, secret affair.' The cards dictated that their love would not only be very intense, but further described that the symbols read a 'secretive' and 'defensive' nature from Gundham in reference to Tanaka's supposed fling with the inanimate object, implying that perhaps he did not like to be accused of having a love or obsession with such an object. This second stream only fueled the ship further, not only spreading awareness for the ship, but causing more fans to consider the powerful dynamic existing between the duo.
Also, in trial 4, he actually makes love to glue. According to him, "Glue... means more to me than even my devas..." implying that he cares for glue more than anything else... more than life itself...