The short answer to the above question is “no,” but there are some caveats that we need to discuss (otherwise, this would be my shortest blog post ever).
Limited liability companies do not require ownership to be evidenced by physical certificates, though a company’s operating agreement can provide, voluntarily, that certificates will be issued. More often than not, in my experience, ownership in LLCs is set forth in a table attached to the operating agreement that is updated as ownership changes. This table can reflect either share-like units of ownership called, appropriately, “units” or percentage ownership. The latter is more unwieldy, I think, particularly with LLCs with many members, but you see it a lot particularly in more old fashioned forms.
For privately-held corporations, most states now permit the issuance of “uncertificated” shares, meaning as with LLCs that there is no physical certificate issued and the corporate records must reflect current ownership. Both New York and Delaware permit the issuance of uncertificated shares by resolution of the Board of Directors (Section 508 of the NYBCL and Section 158 of the DGCL, respectively). In some cases, the corporation will send a notice of issuance of stock to the holder. This open source form from Orrick resembles an actual stock certificate in some ways – it’s not really necessary to do it this way, but it may be more reassuring to old school investors who aren’t aware of the trend toward (and legality of) uncertificated shares.
Finally, there are public companies (usually corporations), where uncertificated shares have been more prevalent for a longer period. This SEC summary outlines the possibilities – if you don’t hold a physical certificate, you either have “street name” registration or “direct” registration (DRS). Since 2008, all public companies with stock listed on a major exchange have been required to have DRS-eligible shares.