Does BCH have Ordinals / Inscriptions / BRC-20 tokens like Bitcoin BTC?
No, but it has CashTokens which is way better.
The accidental discovery of a Taproot exploit allowing Ordinals, Inscriptions & BRC-20 tokens on BTC has created enormous confusion and controversy within its own community. The BCH community resisted the prerequisite implementation of SegWit (it was one of the contentious issues in the Blocksize War) and split away at the same moment SegWit was integrated into BTC. BCH also does not have the Taproot ""upgrade"" added to BTC in 2021. These changes were rejected by the BCH community for their poor engineering choices, which in hindsight has proven very wise.
Instead, the BCH community have deliberately developed a superior system called CashTokens which covers much of the same market requirements (plus a LOT more!) without technical workarounds or community contention.
For extensive discussion & coverage of the Ordinals mess on BTC, see the following Podcast episodes:
Episode 71, shortly after the introduction of Ordinals.
Follow on discussion in Episode 80.
See also: How does Bitcoin Cash governance / updates work?