Dad has a substantial stock portfolio. Can an assisted living facility or nursing home charge him more because of that?

No - they are not income-based - but what can happen is this:

If your dad ends up needing full-time nursing care, and has to go into a nursing home, he will be required to private pay before being qualified for Medicaid - which means all of his stock portfolio and assets will have to be spent down to a very low level (generally $2,000 a month plus one home and one car) before qualifying for Medicaid coverage. And Medicaid looks back 5 years to see if any assets have been divested in that time - and if they have, there's a penalty period equal to the amount that was divested. (So if he divested $30k in assets and then had to go on Medicaid, he would be denied and be required to private pay $30k before he could re-apply for Medicaid.)

Medicaid does not pay for Assisted Living, so he would have to private pay for that anyway.