Flight Instructor with student reported an NMAC with an aircraft in the traffic pattern of N94 non-towered airport. Flight Instructor decided to leave the pattern after the near miss.

Narrative

I am a CFI-I and was performing pattern work at N94 with a student beginning at XA:30 in Aircraft X. After one lap in the pattern; we overheard on the CTAF of Aircraft Y inbound approx. 10 NM east of the field with intention of landing. Over the frequency I chimed in and told Aircraft Y we would perform a shorter pattern so we could give them some space for the approach into N94. Aircraft Y acknowledged the call. After another landing and subsequent radio calls and pattern turns; we finally had visual of the pilot on a 2 NM final while on the downwind leg of runway 28. We extended our downwind to accommodate Aircraft Y. Once abeam of Aircraft Y; we noticed they were at the pattern altitude of 1;500ft MSL and announcing a low approach with intention of staying in the pattern for a landing. This was the first time we heard over the CTAF of any low approach from Aircraft Y; so we had no idea if they were performing GPS-B into the airspace or not. We watched them continue on the upwind and turned base for RWY 28 when we were both beyond the 45 degrees off our left shoulder of the runway numbers and 45 degrees off our left shoulder from Aircraft Y. We continued and did not hear any announcements from Aircraft Y. We performed a touch and go; climbed and began to climb while making calls on the CTAF frequency. We then heard Aircraft Y making the call of a go around. The Aircraft X is a high wing plane so we could not see through the metal wing above us. While the Aircraft Y is a low wing and cannot see through the metal themselves. We were less than 300ft vertical distance from each other once the Aircraft Y was in our view. With the Aircraft Y at a higher altitude than us. We leveled off at 1000ft MSL/500ft AGL while the Aircraft Y continued their turn of upwind to cross wind. We decided at that moment to exit the pattern at 1000 MSL and continue flying away from the airspace.After sometime later; I was able to see the ADS-B tracking thanks to online ADS-B data and saw some errors made. The timestamp over N94 would begin around XB:15. When we were turning our left base; and after the low approach; the Aircraft Y was turning a continuous upwind to downwind while ignoring the x-wind leg. This resulted in their pattern to become extremely tight than normal. The Aircraft Y continued their pattern as normal without extending downwind for us; as if we were not in the pattern or landing. We don't know if we were hidden by the wing of the Aircraft Y; or if the pilot was not listening over the radio; or if this was how the pilot normally treats entering the pattern at N94.