I've found I couldn't set a status to only load on the player or only load on the pet. It would be nice to have a status load condition for "Unit is Unit" where you can select "player" or "pet".

The Unit Type condition already allows to filter by pets or playesr headers.

UnitIsUnit takes other units as well which can also match, like "target", "focus", "targettarget", "focustarget", "arena1", "arena1target", "party1", "party1target", etc. It would be useful to be able to show particular statuses depending on the unit. My request is mainly for the "player" unit though, which is myself in particular.

Players usually don't track their specific things on the small raid frame unit, and for specific class buffs/debuffs the mine filter should work.
But it has sense to display extra/different things in your own separate player unit, and this was implemented at indicator level, in the load condition of each indicators there is an unit filter option, there you can choose to create some indicators only for player, focus, target,targettarget etc frames for example.
In the indicators load tab the name of the filter is Unit Filter, but this filter does not really check for units, it checks for the special units headers that can be displayed detached from the raid group, like targetoftarget, focus, player, etc.
For example maybe you only want power bars for the separate player/target/focus frame units, but not for raid units, so you can setup the unit type load condition to only create the power bars for the separate player/target, etc units.
This filter allows to use custom types, if a custom layout is created in the layout editor, there is a editbox to change the header type name, and those custom header types can be selected in the indicators unit filter too.

I do look at my player frame in the raid frames, especially in arena. I don't show the player frame and only use the raid frame from Grid2 to see myself, so it's useful for me to have player-specific statuses and information there. I don't want my player frame duplicated somewhere else, and I use FrameSort to make sure my player frame is always at the bottom. This all works well, except it's missing a way to show self-specific statuses with a "unit is unit" load filter. I've also wanted to attach TellMeWhen icons to my specific player frame, and that would work with a "unit is unit" load filter.

I could think on adding a UnitIsPlayer filter but not a generic UnitIsUnit filter, because it is too resource expensive (all filters in all units should be refreshed too often in combat: with every target/focus/targettarget,... change).

That would definitely be useful for me, so if that's something you're willing to add, I'd be very happy with that. It's not the full request, but having that would be great at least.

Thank you for your implementation. It works for 95% of what I was interested in using my request for, and anything beyond that may be only a luxury which is more convoluted than it's worth. As a result, I'm closing this.

"Unit is my pet" might be not too far off and useful in a raid scenario or battlegrounds with multiple pet classes, but it's not important enough for me to leave this issue ticket hanging open.