Based on what is written in the Stata 17 Manual pp 60-66 and documentation for pdslasso, they are exactly the same (although dsregress has richer set of options).
- Both documentations reference Belloni et al. 2014 paper
- Both documentations show basically the same model (Stata manual transposes beta vector instead of x matrix but once you go through the math the expressions are equivalent as far as I can see).
Next, this would not be the first time when there already was user written package for Stata, and later when new version of Stata is released developer just add renamed version of that package. Stata is developed for profit, so I guess it makes sense to do things twice so they can sell users the "new Stata x.x" with "new" features like lasso.
Stata added lasso models, including dsregress close to the end of 2019 in Stata 16, the pdlasso package was out in late 2018/early 2019.