TRACON Controller reported two separate instances of aircraft descending below the published altitudes of the same instrument approach and flying below the minimum vectoring altitude.

Narrative

Aircraft X on the approach into ZZZ was at 128 MSL 1.5 miles from ZZZZZ. ZZZZZ crossing altitude is 129 MSL. Approach issued safety alert and Pilot reported 'I almost got the field in sight.' Over ZZZZZ1; the Pilot was at 121 MSL. Crossing altitude for ZZZZZ1 is 123 MSL. I gave an advisory since the Pilot was already passed the fix. Pilot stated; 'I can just barely see it.' Aircraft Y; also cleared the approach; checked on 1 mile from ZZZZZ at 126 MSL. Safety alert was issued and no response was received. Aircraft Y continued his decent. .5 mile from ZZZZZ second safety alert was issued when Aircraft Y was at 124 MSL. Aircraft Y crossed ZZZZZ at 122 MSL; below the crossing altitude ZZZZZ1. 2 safety alerts issued and the Pilot never responded. Pilot was advised of crossing altitudes; no corrections were made. I took the next appropriate action and issued a go-around. Pilot responded confused crossing ZZZZZ1 at 112 MSL. The wrong published altitude for the approach the Pilot was cleared on. Never read back 'go-around.' a second go-around was issued at which point the Pilot began to comply. Pilot cleared approach advised me during clarification that they were following 'company visual approach;' which they were never cleared. Nothing to recommend. Controllers should be alleviated from any wrongdoing when it comes to a pilot accident in the future due to improper pilot compliance. Its been reported time and time again.