Most times but not always unfortunately. Had a couple of times that I did not check a small frontend change on firefox because I assumed a small change would be fine, yet had different results in chrome as on firefox.
I did not check on mobile though.
I use Firefox on Android and often run into web sites that don't work right because they are dependent on some Chrome or Webkit behaviour, and I need to switch the browser to desktop mode to resolve the problem.
It's rare for me to find one, although the highest profile annoyance for me is textboxes break on m.facebook.com. The caret refuses to move from the current line without deleting or adding a character.
Known issue and I'm yet to see a fix (other than "mbasic" which you have to use for messenger anyway).
Almost everything else seems to be fixed with uBlock Origin and/or Reading Mode.
I experianced this myself and was pretty sure this is on purpose. If you request desktop on Chrome for Android you get a stretched out version of the mobile site and the auto complete of your keyboard makes the textbox on messages impossible to use. I think they want to funnel people to the app I just don't go on the site anymore.
i often run into google web sites that render and behave terribly in android chrome (usually floating windows with the close hidden or extending outside the screen).
most web problems are now bugs either by the site developer or in a specific browser.
thats a long way from ie6 problems where either you needed a unique site configuration for every browser or supported ie6 only.
The other thing is that major libraries (CSS, JS: jQuery, bootstrap, React) make sure they work on all browsers so you don't have to worry about that part.
The winXP-IE6 was hell because that never happened, you always had to changes to make it work on IE6 and some other changes to support IE7 and so on.