Although the plumber said after looking at report pictures, he thinks it is terra cotta.

Its terra cotta pipe , its old . 

HiCurrently live in a 1958 home with an oil furnace from 1997 and original terra cotta tile lined flue, which I am told would be a good idea to have relined with a stainless steel liner because the tiles are no longer aligned.

Even low-fire terra cotta work usually fires at 1945F.

If it is a more porous clay/terra cotta style tile without glazing you can typically use any kind of concrete type stain or paint to color it. 

@John WarrenSeparate roofing but they partially share a parapet wall with terracotta coping on top.

I also see an emphasis on natural design elements (clay, linen, wood, terracotta, ), and lots of plants.

It could even be as simple as a terra cotta pot with a recirculating pump from the hardware store ($50).

Sometimes companies that provide chimney liners try to get you to add metal liners to chimneys that already have a terra cotta liner in good condition which isn't necessary.

My house is historic and terra cotta pipe still in places.