Both Altera and Xilinx were on TSMC. Altera wanted an Edge over Xilinx, at the time Intel was committing ( on paper ) to their Custom Foundry. Altera switched and bet to Intel Custom Foundry. Nothing ever worked out with Intel Custom Foundry because they were not used to working with others on Foundry Process. Intel thought the problem was with Altera not being part of company and they had too much cash so they might as well buy them for better synergy. And it did, getting internal access seems to have ( on paper or slides ) speed things up with product launches and roadmap, until they hit the Intel 10nm fiasco.

I believe Altera was already manufacturing chips using Intel process prior to acquisition, while Xilinx is using TSMC?

Altera Stratix V FPGAs actually had more market share than Virtex 7s. They were better chips. That said, the production delays around Arria 10 and Stratix 10 and the time lag caused by the Intel acquisition totally killed their market position. The only reasons to use Intel FPGAs now are (1) 64-bit floating point support or (2) if your Intel salesman gives you a really good deal.

It may have been Xilinx not wanting to get into bed with Intel. Xilinx may have wanted a degree of technical independence or freedom to carry out their own strategy that was not forthcoming from Intel.