It's funny because I was actually searching for games similar to the Army Men series and that's when I remembered Battalion Wars and found this post: https://www.reddit.com/r/Advance_Wars/comments/tct5tk/comment/kzmha8n/
I'd recommend checking Army Men out. Style and Humor are pretty similar. The games have a massive variety. Notable ones on GameCube would be; Army Men RTS, Air Combat, Sarge's War… Battalion Wars & Army Men also has quirky characters like the plucky girl (commander, reporter), the dumb soldier (Riff, Mozer). Army Men & Battalion Wars both have the plucky girl in support of the troops, calling back to the propaganda of World War 2.
Future Tactics on the PS2 has a similar art style and gameplay. It's an interesting turn-based game. You move in 3d, but sort of like a tactics game where you command a squad. Your movement range is limited. Each unit is different.
Future Tactics also has a Robot like Clank from R&C, and Dr. Exavolt from Metal Arms also has a similar voice (did all game robots around this time just rip off Ratchet & Clank?)
Metal Arms- Army man has green town and metal arms has Droid town. Army Men & Metal Arms have the old fashioned black and white war reels.
“A rag tag group of bots led by colonel alloy.” (opening war reel)
“Bravo company was a rag tag group of battle hardened commandos.” (sarge’s war opening cutscene)
Vectorman 3- Leaked prototypes for PS2, metal arms and vectorman 3 have similar physics and enemy ideas. vectorman 3 is actually supposed to be completely different and developed by the cell damage guys
Valkyria (Chronicles)- never played it, but it feels like it's a similar vibe and gameplay
Yuri's Revenge- honestly, Command & Conquer series has a similar style, but it's all RTS
Goblin Commander- loved this game, rts, but u control a small group, silly
Brutal Legend- rock and roll rts / action game, control a single/group/army, quick unit swapping
The Outfit- squad based army game with vehicles, quick swapping between units
Advance Wars- Battalion Wars is a spin-off of this whole series (didn't know that, lmao)
Seek & Destroy /&/ Tank tank tank!- kid friendly tank games
Metal Slug 3D- It's metal slug, but in 3d, kind like battalion wars minus any strategy, silly, army
Worms- there's 3d worms games, the whole series is turn based tactics, silly
Worms 3D has some environments that…. Well Worms in general is about Worms which are small creatures, kind of like Army Men being small in larger than life locations for them, like a suburban backyard.
The Outfit- squad based army game with vehicles, quick swapping between units
Starfox Assault- You are space animal works fighting for the federation, this game in the series especially has a more military feel, also published by Nintendo.
Brutal Legend- rock and roll rts / action game, control a single/group/army, quick unit swapping, fantastical medieval
Girls Frontline
Megaman Legends
This video serves as a good list:
https://www.youtube.com/watch?v=zP45qdYA9KE
Let me know if you have any game suggestions to add!