I think it depends on the size of the TV and how much actual room you have because both of your options seem to need a complete wall to be accomplished.

It looks like you have a radiator that would limit the built in look. Personally, I'd just get a free standing entertainment unit.

I like Besta from Ikea you can really customize to exactly what you need and IMO TV are usually hung too high. You want a cabinet that keeps the TV center at your eye level when you are seated

It really depends on your style. The built-in looks traditional with the molding and even a bit farmhouse-y with that shiplap wall. The other is more mid-century modern with a hint of industrial. Do you have a better photo of the room from farther back you can post? Based off the blue crush sofa, pendant light shadow form and gold accent table, my gut says that the metallic bookshelves with floating console would fit your style best, but again, just a guess off very little information provided.

Here is a photo from further back