I am assuming pets work in Arena? If not, disregard this topic.
Assuming pets work in Arena: it’d be nice to to click the opposing Hero’s portrait to see what pet I’m facing (every bit of info gleaned helps, especially in Arena). I shouldn’t have to go to an outside website to match the pet’s likeness against 30+ pets and figure out what the bonus is.
Again, if pets work in Arena 
I’m pretty sure, just like with traits and guild bonuses, pet bonuses don’t affect the arena. Even if they did, just seeing the pet they have equipped wouldn’t do much good. Pet bonuses stack, all the bonuses you get are active regardless of if they’re visually equipped. You’d need to see their entire inventory to know what all their bonuses are. That seems a little over the top. If there’s a team bonus, you’ll see elevated stats, no need to see their pets. We never see their mastery and it doesn’t really make a difference. Gold, souls, and xp are totally irrelevant to the fight. So no need to see all their pets in any situation.
Oh right, can’t tell a level 1 pet from a level 20 pet. Okay then.
You don’t need to see a level, their stats will reflect what bonuses they have. It’s just like you don’t “see” exactly what bonuses a person has for using an all blue team, or all fey. Plus they aren’t just getting a bonus from that one visually equipped pet. Every single pet they own gives a bonus when the team conditions are met. Say I’m using an all blue fey team and i have grimlet equipped visually. His bonus will not work with that team, he’s just there to look cute. I could have crabbie or feyrie, whose bonuses would apply, but you just can’t see them. If i had both, i would get the bonuses from both of them for the team even though neither were visually equipped. Do you need to see that i have them? No. My stats would just be slightly higher. And that only matters for pvp anyway. In the arena, none of your bonuses apply. It’s your basic hero stats. You don’t get guild bonuses, so you shouldn’t get pet ones either.
You missed the important part.
The Equipped pet is cosmetic. When pet bonuses do apply, all pet bonuses apply additively.
If I have Lucky equipped, my Minishroom bonus is still active. If I have no pet equipped, I still have all of the same bonuses.
So it doesn’t matter what pet your opponent has equipped. Also I’d be surprised and upset to find out pet bonuses work in Arena. It’s supposed to be a mode where “outside stuff” doesn’t work.
Speaking of “outside stuff,” doesn’t the Hero class have some effect? Not the traits per se, but stuff like "counts as X (Mech/Knight/etc) with regards to spells?
Huh; if all pet bonuses do apply additively, in that case, I’d like it so if I click the opposing Hero’s portrait, maybe I get to see what pet bonuses I’m up against? I guess now I’m talking about PvP.
And another thing; are pet bonuses active even if Hero isn’t one of the four troops being used?
Yes, he is.