I had a problem using the calendar on an Android phone while using Chrome.
I already solved it myself, but you might be interested in knowing the details:

While using the "Embedded javascript calendar", clicking the dates didn't call the "onupdate" function, as it should.
Switching to the regular "Javascript calendar as a plugin" calendar, clicking on the input field didn't open the calendar popup.

I noticed jsuites.net didn't have this problem, so upon doing some testing I noticed that adding a script that jsuites.net has ("[email protected]/Tocca.min.js") on my website, solved the problem.
Don't know why.
If this script IS required, you should put that information on the website.

I am 90% sure this wasn't a problem a few months ago... I'm assuming either chrome or android or your code changed somehow during this period of time and started having issues.

For clarification:
The calendar had no issues when I tested it on desktop
The calendar had no issues when I tested it on iPhone while using chrome
The calendar had no issues when I tested it on tested on android with other browsers (the default one and yandex)

Only had issues on android + chrome