IRS Interpretation of the Calendar for Long-Term Capital Gains Purposes

From IRS Publication 544 (2024), Sales and Other Dispositions of Assets:

Long and Short Term

Where you report a capital gain or loss depends on how long you own the asset before you sell or exchange it. The time you own an asset before disposing of it is the holding period.

If you received a Form 1099-B (or substitute statement), box 2 may help you determine whether the gain or loss is short term or long term.

Generally, if you hold a capital asset 1 year or less, the gain or loss from its disposition is short term. Report it in Part I of Form 8949 and/or Schedule D, as applicable. If you hold a capital asset longer than 1 year, the gain or loss from its disposition is generally long term. Report it in Part II of Form 8949 and/or Schedule D, as applicable.

Now they discuss the math:

Holding period.

To figure if you held property longer than 1 year, start counting on the day following the day you acquired the property. The day you disposed of the property is part of your holding period.

Example.

If you bought an asset on June 15, 2023, you should start counting on June 16, 2023. If you sold the asset on June 15, 2024, your holding period is not longer than 1 year, but if you sold it on June 17, 2024, your holding period is longer than 1 year.

Using the dates you included in the question:

a stock is purchased on 8/21/2023 and sold on 8/21/2024,

So you start counting on 8/22/2023, if you sell on on 8/23/2024 you have held it for more than a year.