If it were salvage that'd just mean they have the right to receive a proportion of fair value for recovering it, not the right to do anything with it.
Maybe SpaceX were aware the Russian would want to steal it, so the US government deliberately created the fire in the radar to force a delay that would allow SpaceX time to come up with something that would give the Russian invalid information when they recovered the stage.
Here's a relevant comment posted on the NasaSpaceFlight forum a few days ago:
This Russian tug, as best I can determine from the Marine tracking sites, wasn't anywhere near Cape Canaveral (or Georgia/S. Carolina where it would had to have been to see the first stage reentry) when the Falcon 9 launch occurred yesterday. It was off Miami, headed south.
But, alas, since it was reported in the news and since Elon linked to the report in a tweet, it will forever incorrectly be recorded in history that Russia sent a ship to spy on a SpaceX launch.
Other posts there note that the Russian ship may have been there to monitor the earlier NRO reconnaissance i.e. spy satellite launch (or at least to annoy the US as it launched its spy satellite), and had little/no interest in the SpaceX launch.
So enough about the Russian ship already... ;-)