in your provided JSFiddle, you have following code
var tooltipContent = $(this).attr('data-ttl');
var newElem = '<span class="tooltip">'+ tooltipContent +'</span>';
instead of adding title, try to add following data attribute in your P Tag
<p id="p1" data-title="tooltip 1">Go</a>">
This is a paragraph
</p>
<p id="p2" data-title="tooltip 2">
This is another paragraph
</p>
and change code to following
var tooltipContent = $(this).data('data-title');
var newElem = '<span class="tooltip">'+ tooltipContent +'</span>';
Please find updated JS Fiddle
EDIT
If you check the actual jQuery UI tooltip, default browser tooltip doesn't appear. jQuery UI Tooltip
The reason is that, if you check HTML in Developer tool window you can see the library itself remove content in "title" attribute on mouseenter and reset it again on mouseout. After looking at your fiddle, it seems you are adding your own logic to show and hide tooltip. Well incase you add your custom logic you have to handle those all cases as well. Do following step to have same behavior as default jQuery tooltip.
Remove content in the "title" attribute when you create tooltip and show (on mouse enter).
Store that content some where or in "data" attribute of tag.
Reset the content again to "title" attribute when user moves out the mouse from the control.
Please refer this JS Fiddle with solution for your issue. i have modified your tooltip click event to restore the title content
Updated JS Fiddle