I noticed that in the Tridentine Missal certain days have more than one Mass for a Saint.
e.g. on Jan. 18 is the feast day of St. Peter’s Chair at Rome which is a double major and certainly takes preference. But after the propers for that mass shoes St. Prisca, Virgin, Martyr which is not as important feast day.
Now couldn’t the same be done with St. Linus and St. Padre Pio on Sept. 23
Are you quoting from a pre-1962 missal? According to those rubrics, in some places, on occasion of festivity two conventual Masses were allowed. Common practise with two feasts or ferias with proper Masses (like in Lent) coinciding was celebrate one of higher rank and commemorate the second.
So for example if St. Thomas Aquinas fell in Lent the practise would be before the changes of 1955, to say his Mass but add the collect, secret, postcommunion of the feria and say as the Last Gospel the Gospel of the feria.
However, two conventual Masses could be said: one of St. Thomas and the other of the feria without the commemorations.
Though I’m not fully sure St. Prisca was an applicable situation. In any case, her collect would be given on account of commemoration and also a Mass given (or a direction to the Common) in case she was “bumped up” to a Double of the I or II Class.
I’m pretty sure that two conventual Masses would normally also not be allowed according to the pre-1955 rubrics for the sitaution with St. Pio.
However, according to the 1962 rubrics, there is only usually one conventual Mass. And ordinary commemorations, etc. are curtailed.
There is a difference, also, between celebrating a feast if it is inscribed in the local calendar and if one is celebrating it by virtue of it occurring in the martyrology at that date. If St. Pio was inscribed in a local calendar as a particular feast of the III Class, his Mass would be able to be celebrated on September 23.
If celebrated by virtue of it occurring on that date in the martyrology, it can only be celebrated on a day of the IV Class which would be, as AlexV pointed out, September 24, barring it is a Sunday or Ember day