Credit unions are subject to essentially the same capitalization requirements as banks---they're regulated by NCUA, but NCUA's capitalization requirements are largely harmonized with those of other bank regulators. It's important to understand that banks themselves are not all regulated by the same agency, with bank regulation split between FDIC, OCC, and the FRS depending on the bank (this is mostly unrelated to FDIC insurance which applies to depository financial institutions regulated by OCC and the FRS as well). These different regulators all apply somewhat different supervision methodologies, with the result that banks do indeed "shop around" for a regulator that they feel will work the best as a long-term relationship. But the overall capitalization ratio requirements are mostly the same at 7-10% being well-capitalized depending on calculation method.
But this discussion is more about insurance, not capital reserves. Credit unions are required to hold insurance (coverage) on their deposits just like banks, with the same cap of $250,000 per account. Most, but not all, credit unions are insured by NCUA, backed by the US government. All federally chartered credit unions are insured by NCUA, but state-chartered credit unions are not necessarily required to be. The majority of state-chartered credit unions are also insured by NCUA, but they have the option of obtaining their insurance by other means, and some choose to use private insurers like American Share Insurance. These private insurers are usually backed by a huge reinsurer and so the risk of them not meeting their obligations is low, but arguably higher than NCUA. On the flipside, private share insurers sometimes offer higher coverage limits than NCUA. It's mostly a minor issue though as credit unions covered by other than NCUA are uncommon, and NCUA-insured credit unions prominently post the NCUA logo. Similarly, non-NCUA credit unions are required to disclose their insurer.
Federally-chartered credit unions usually use "Federal" in their name although some don't use it in their general advertising and logotype any more. State-chartered credit unions only exist in some states, but California charters credit unions and as you'd imagine there are quite a few examples in that state. There are even "dual-chartered" credit unions in some states that hold charters from both state and federal governments. This is the norm in e.g. Washington due to some banking regulation history. Older credit unions are more likely to be state-chartered as the federal system is newer than most state systems, but credit unions didn't really take off until the Federal Credit Union Act so there's still not that many of them.