There's a lot to un-pack here....
First off, opening a link with target="_blank" or opening a link manually as a new tab should pick up the referrer the same.. but this is all dependent on if there are tags in the site that are actively blocking the passing of referrers (i.e. rel="noreferrer")
Referral logic is based on HTML headers, so as long as the headers are working properly then Adobe Analytics (and other web analytics tools) should all be able to read the information properly).
Now, Referrer information is used in the Referrers Report (and when an internal referrer, the value is ignored/not included in the report). But referrer data is not used for flow diagrams (page to page to page) that is based solely on hit timestamp data.
Now.. when it comes to "engaged time"... this is always going to be an imperfect solution, since Adobe is not a heartbeat system. Time spent is based on timestamp differences between page views.... Time Stamp B - Time Stamp A....
If you are navigating in multiple tabs, the math for the time spent will not be isolated to individual tabs, As soon as Adobe sees a new page view (no matter which tab) that will be used to calculate the time on the last page viewed...
The last page viewed in your site (and subsequently single page visits) will not have any time spent associated to that view, as there is nothing to calculate the time spent against.
If you want more accurate time spend, you would be better off getting another tool (on top of Adobe) that uses heartbeats.. most of these also take into account if the tab is active or not.
The time spent metrics in Adobe are good for comparisons over time, are people spending more or less time on the site, or per page, etc... but its not good for accurate time spend...