Yes, but it was a little bit more complicated than that.

There were certainly widespread formal uses of captured enemy tanks in WWII. The Germans captured a vast quantity of French and British tanks in 1940 that were considered worthy for service, either as training or front line vehicles. Some went into battle largely as is, like the SOMUA S 35 tank or the Cruiser Tank Mk.IV (Kreuzer Panzerkampfwagen Mk IV 744 (e) in German parlance), which can be spotted in photos taken in June 1940 with German markings hurriedly slapped over top of the British ones. After the fighting ended, captured tanks were sifted through and used for training or sent to the East, depending on whether they were considered fit for battle or not. Some were also used as a chassis for artillery tractors and SPGs.

The pattern didn't change drastically with the invasion of the USSR: captured tanks were used often. T-34s were the most popular, and they usually had a commander's cupola from a PzIII or PzII installed. There is also a famous photo of a KV-1 with a 75 mm cannon and a cupola installed. All of these tanks were also painted in German camo and with huge Balkenkreuze symbols painted on every surface to prevent friendly fire.

The situation on the other side of the front was the same. Plagued by shortages of tanks while factories were setting up in the east after being moved from occupied territories, the Soviets needed a fresh boost to their tank armies, so captured tanks went into action, chiefly the PzIII, PzIV, and StuG tanks. The tanks wouldn't just be captured and used as is, they would be collected and sent to the rear to be refurbished. PzIII and StuG chassis were also used to make SU-76I and SG-122 SPGs, which resolved supply issues with German ammunition.

As the war went on, issues with tank production improved, and there was no longer a need to scrounge up German chassis, so the conversion program stopped. Use of captured tanks declined drastically, partially because newer German tanks took expensive high octane gasoline to run (by this point Soviet and Lend Lease tanks used by the Red Army ran chiefly on diesel fuel), and partially because they were very unreliable. For instance, two captured Tigers that the 3rd Guards Tank Army attempted to use seized up during a 150 km march and had to be abandoned. In the fall of 1944 the army reported that it had not used any captured vehicles in combat at all in 1943 or 1944, citing lack of fuel and trained crews. One Tiger that was bogged down in a swamp was recovered by the 28th Guards Tank Brigade and went into battle on January 5th, 1944, but was quickly abandoned because it broke down. The same situation repeated itself a handful of times in 1944: a Tiger is captured, fights for a few days, then is abandoned. Correspondence from repair yards complaining about a lack of spare Tiger parts indicates that an attempt was made to repair them, but it's impossible to say if any were actually restored to fighting condition and used again.

The situation with Panthers is similar. Individual brief instances of use in combat can be spotted starting in the fall of 1943, but the tanks broke down very quickly. Like the Tigers, Panthers were sent to Soviet repair factories, but lack of spare parts made putting them back into service difficult.

Overall, just ditching your own tank for another one would have been a pretty tall order. Keep in mind that the tank commander has his objective, and mucking about with a random tank you find on the battlefield is not a part of it. A tank that is abandoned could be put out of action, booby trapped, simply stuck, low on fuel or ammunition, etc. The best you could do is report this tank's location so it could be checked over and evacuated by your company workshop, and then if it still runs then it would be very possible to squeeze a few more battles out of it.

Y. Pasholok Big Frenchman with a Balkenkreuz

Y. Pasholok A Bit More Armour

Y. Pasholok Conditionally Viable

P. Samsonov Fate of Baryshev's tank

Y. Pasholok Assault Gun on a Foreign Chassis

P. Samsonov SU-76I

P. Samsonov Valentines and Panthers

P. Samsonov Trophies in Combat

Y. Pasholok Heavy Trophy

Y. Pasholok No Beast More Fearsome than a Cat

P. Samsonov Captured German Tanks in the Red Army