Yes, they do. It's a flat damage multiplier, not physical attack stat or anything wonky like that. So do your enemies, though, so be careful.
Some other notes:
I'm not sure if it's intentional, but as of 2.2.500 at least, combat drops do not get multiplied by the x25 mod (items requiring horror drops to upgrade are gonna take the same amount of time). Neither do Another Dungeon rewards (so no easy 25 Treatises). For episode items, I haven't checked, but I'm not sure it's a good idea to have x25 on there - the only place I can think of other than monster drops to get them is chests, and they usually come in numbers larger than 25.
If you haven't done it yet, do the Persona 5 side quest. Use a guide if you need to, but make sure you get the x25 for the badges that drop during the episode. Avoid it on the personal weapons, though - no point in having 25 copies of a weapon you can't sell, break down, or use on anyone else.
For the Cruel Angel gear in Konium, you can absolutely get all the items by having x25 on when you pick up one each of the items. You can also get 25 of each Cruel Angel gear by keeping it on while exchanging, which is useful...with the downside that each piece is gonna take a long time to upgrade due to scarcity of the horror mob drop required to upgrade, so keep in mind most of them will not be upgraded.
[Edit:] If the game goes idle and the screen locks, sometimes the BMT menu disappears (like when it's hidden with the 'hide' feature). Is there some way I haven't found to get it back without closing and reopening the app entirely?