Do standard NiMh AA or AAA battery's have any kind of BMS or protection circuit built-in?
No, standard, "off the shelf" NiMH batteries have no electronics or monitoring system.

If you want a circuit to prevent draining low batteries, look for UVLO or Under Voltage circuits or ICs.

https://www.mouser.com/ds/2/268/50002561A-1102171.pdf is an application sheet for a MCP16251 One-Cell Boost Converter with an External UVLO Circuit, compromised of basic transistors and resistors. The MCP16251 can also boost 2 NiMH cells up to 5V if needed. But there are other dedicated UVLO ICs or NiMH.

Then again, NiMH doesn't suffer catastrophic damage if you try to drain them too much, unlike Lithium cells, so you could skip this if you want.