In a touching tribute, Mick Schumacher presented Hamilton with one of his father's race helmets after he equalled the senior Schumacher's 91 wins.
The positive test is the first by a team member at a race weekend since Sergio Perez of Racing Point returned from Mexico ahead of the British Grand Prix in July.
Victory for Hamilton at the Sochi Olympic Park would see him draw level with Michael Schumacher on 91 career wins, more than any other driver in the 70-year history of F1.
Sergio Perez will be replaced next season at Racing Point by Sebastian Vettel.
FIA investigating Mercedes' Lewis Hamilton for Breonna Taylor t-shirt at Tuscan Grand Prix, says report
Lewis Hamilton, who won the Tuscan GP, wore a t-shirt bearing a message that said "Arrest the cops who killed Breonna Taylor" before and after the race, during an anti-racism ceremony and television interviews.
Ferrari reverted back to an age-old burgundy livery for its 1,000th F1 race. But there was little to celebrate as Charles Leclerc finished eighth and Sebastian Vettel 10th.
Formula 1 2020: Contract talks between Mercedes and Lewis Hamilton still 'a work in progress', says Toto Wolff
Hamilton had said in July that he expected it to be a relatively straightforward process and "not a big effort" and he had no desire to move to another team.
Formula 1 2020: Ferrari chief Mattia Binotto says team looking to put 'worst of days' behind at Mugello
Ferrari also finished outside the points in the previous race in Belgium and the second successive double flop dropped them to sixth in the constructors' championship.
Formula 1 2020: Pierre Gasly races to stunning victory at Italian GP as Lewis Hamilton handed penalty
Pierre Gasly finished 0.415 seconds ahead of McLaren driver Carlos Sainz and 3.358 ahead of Racing Point's Lance Stroll on a surreal-looking podium.
Formula 1 2020: Ferrari hurting after both drivers fail to qualify inside top 10 at Italian GP since 1984
After their pointless flop at the Belgian Grand Prix last Sunday, this was another embarrassment for the 'scarlet scuderia', but little surprise to their drivers.
Formula 1 2020: Eight drivers summoned by stewards over Lewis Hamilton near-miss during Italian GP practice
As a result of the incident, when Hamilton changed direction at around 300 kph, he was summoned along with Nicholas Latifi, Romain Grosjean, Sergio Perez, Pierre Gasly, Lance Stroll, Alex Albon and Carlos Sainz to explain what happened.
Defending his decision to fly to Mexico, Perez said: "I think I was just unlucky. I haven't done anything different to the rest of the paddock. I'm not willing to take any blame for that because anyone can get it."
The Racing Point car has been dubbed the "pink Mercedes" because the team is accused of copying the part from Mercedes.
Formula 1 2020: Mercedes' Lewis Hamilton compares Pirelli tyres to balloons after blistering problems
Hamilton said he was glad to have survived the race without a puncture and to have found some pace to claim second at the finish.
Formula 1 2020: Valtteri Bottas seizes pole position for 70th Anniversary Grand Prix ahead of Lewis Hamilton
This is Valtteri Bottas' first pole since the opening race of the pandemic-delayed season in Austria, which he went on to win.
Formula 1 2020: Mercedes' Lewis Hamilton fastest ahead of 70th Anniversary Grand Prix qualifying at Silverstone
Lewis Hamilton holds a 30-point championship lead over teammate Bottas in his quest for a record-equaling seventh world title.
Formula 1 2020: Lance Stroll quickest at 2nd British Grand Prix practice; Sebastian Vettel struggles
Lance Stroll's unexpected teammate Nico Hulkenberg was seventh after filling in at the last minute for Sergio Perez.
According to the media report, Sebastian Vettel is expected to replace Sergio Perez, who currently races for Racing Point and has a lengthy contract with the team.
Hamilton took a knee with 11 drivers before the at the Styrian GP while wearing a Black Lives Matter T-shirt and then gave a raised fist salute on his car and on the podium after winning.
Formula 1: Charles Leclerc apologises to Ferrari for first-lap collision with teammate Sebastian Vettel
Charles Leclerc's car was tipped onto Vettel’s after he tried a lunge down the inside of the German and hit a kerb in the turn three incident.