Does an Intel or AMD motherboard have Intel or AMD SATA controllers?
Yes. The chipset has a number of built-in SATA ports. On Intel motherboards with Intel chipsets, the majority of the SATA ports will be Intel. Same with AMD. Note that there are (or used to be) various 3rd party chipset vendors such as Nvidia (still has some AMD chipsets). Of course even on an AMD based board, if the chipset is Nvidia then the SATA ports will be Nvidia.
Note that chipsets only provide a limited number of ports, and sometimes a motherboard will have more ports provided by a 3rd party, such as Marvell or Jmicron.
For instance, my motherboard is an Asus P8P67 Pro. It has a total of 10 SATA ports. The Intel chipset that the motherboard is based on (P67) only provides four SATA II ports and two SATA 6G ports for a total of six. Two more SATA 6G internal ports on the board are provided by a Marvell chip, and the two eSATA ports are provided by a Jmicron chip.