Files at geofabrik.de do not have that metadata. You can collect your own metadata by checking also the corresponding web page when you download data. For example the latest data for Andorra is http://download.geofabrik.de/europe/albania-latest.osm.pbf and the web page http://download.geofabrik.de/europe/andorra.html informs
andorra-latest.osm.pbf, suitable for Osmium, Osmosis, imposm,
osm2pgsql, mkgmap, and others. This file was last modified 12 hours
ago and contains all OSM data up to 2020-10-19T20:42:02Z. File size:
1.6 MB; MD5 sum: 879e2fb46dd58f301f18986a1ecb7770.
If you have a pbf or osm-xml file and you do now know the age of it you can only scan the whole file and find the latest timestamp of any node, way, and relation. As you can see from the xml extract from Geofabrik, there are no timestamps about the whole file.
<?xml version='1.0' encoding='UTF-8'?>
<osm version="0.6" generator="osmium/1.8.0">
<bounds minlat="42.4276" minlon="1.412368" maxlat="42.65717" maxlon="1.787481"/>
<node id="625025" version="3" timestamp="2019-12-28T18:39:17Z" lat="42.5142133" lon="1.5527243"/>
<node id="625026" version="2" timestamp="2014-04-16T13:38:15Z" lat="42.5144526" lon="1.5526978"/>
<node id="625027" version="1" timestamp="2006-03-14T13:46:57Z" lat="42.5152372" lon="1.552978"/>
<node id="625028" version="2" timestamp="2014-04-16T13:38:15Z" lat="42.5165425" lon="1.553585"/>
<node id="625029" version="2" timestamp="2014-04-16T13:38:15Z" lat="42.5168939" lon="1.553855"/>
<node id="625030" version="6" timestamp="2018-12-21T11:04:10Z" lat="42.517845" lon="1.5552446"/>
<node id="625032" version="3" timestamp="2020-03-01T20:02:46Z" lat="42.5213647" lon="1.5587682"/>