I have two questions,
1.) Are tagged pets and themed pets the same thing?

2.) If you breed two themed (Tagged?) pets (same theme) will the offspring be themed (tagged?) as well?

To answer your first question, tagged pets and themed pets are usually the same thing. Let's say during a Seasonal Festival, you encounter a themed pet and you trap it. It will have the type of theme it is and the species as a "tag" by its genetic mutations. Here is an example. This is an Emperor Butterfly Morkko I tamed during the last Summer Festival.

https://www.sylestia.com/view/pets/?petid=2616166

Because she is tamed (same goes for exclusive themes), she has a theme "tag" above her genetic mutations. If you click on her link, scroll down on her page and you can see it.

And now to answer your second question. The offspring will be considered a "purebred" (PB, if you see that around the site and wonder what it means) theme. But it will not be "tagged" like its parents. Here is an example of that.

https://www.sylestia.com/view/pets/?petid=2765101

He is a purebred Horned Owl Nephini. When you click his link and scroll down, you will see that he doesn't have a tag. But when you go and look at his parents, they will have a tag because they are both that particular theme that had an offspring.

I hope that helps, and if you have any more questions, feel free to send me a message here :)
https://www.sylestia.com/view/users/?userid=58248

I'd like to add to Articu's explanation that there are also some themed tamed pets that are not tagged. This doesn't happen anymore though and the old ones are rare so it's seldom to run into one. Tamed themed pets that don't have tags are called "pre-tags". These were themed pets released for the 2012 Christmas event, 2013 Valentine event, and the 2013 Spring fest. The tagging system wasn't put into place yet. Tamed themes starting in the summer of 2013 and on forward have tags.