I've been up here for forever and still have no clue what a themed pet is, all I know is that there's something on the pet's profile that says it's themed if it is. Can someone please explain?

This is an Angelfish Aerdini (clickable image). Themes are pets of the same species with the same color set that are obtained during festivals or events, or in the Lost Grove, and first-generation have purple text for their theme after a gold star. If you breed two pets of the same theme, you get a purebred of that theme.

As you can see, (click) this pet's ancestors are all Purebred or Tagged (1st gen) Angelfish Aerdini. If you go to its profile, it will say Angelfish Aerdini in blue text after a blue star. Hovering over the stars on the lineage page also brings up a text that says "Purebred Pet (theme name)" or "Themed Pet (theme name)"
Themes are unique in that the Philter of Themed Trait Disruption, a diamond shop item, allows you to shuffle their traits. This cannot be done with any other kind of pet.
There is often a distinct market for themed pets, as you know the color set and can search for specific themes in Advanced Search.
For example, now is the Week of Love. You can collect Candy Hearts, which when ten are combined, have a chance of giving you an item that allows you to get a 2vis or 1vis of this event's themed pets.
During the seasonal festivals, you can get themed pets two ways. The first way is by planting/brewing/sculpting them in the event area. It's all covered in the Overview. For those pets, you can make them male or female, boost their stats, select slots for their visible traits, of which they can have up to three, and give them recessive genes, all at a price.
The second way is to encounter them in the Lost Grove or event explore zone. When you encounter one, the text will say "You have encountered a Rare Enemy" instead of "You have encountered an Enemy". When you enter the battle, instead of "Wild (species)" It will say "(name of the theme) (species)"
I hope this helps.