Does BCH have Ordinals / Inscriptions / BRC-20 tokens like Bitcoin BTC?
No, but it has CashTokens which is way better.
Technically, Ordinals is just a way of counting satoshis & so is compatible with BCH, but the attached on-chain stored Inscriptions & tokens which made the protocol famous are not. The accidental discovery of a Taproot exploit allowing 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 more about CashTokens, see What is CashTokens?.
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.