Is it really bad to make touch & go's for the engine?

Generally speaking, no. First off "shocking" an engine is generally referred to in the context of cooling it not heating it. For what its worth the consensus is that this is largely an academic debate anyway and may or may not actually contribute to engine wear. There is some logic that dictates that cold oil does not lubricate as well as hot oil but if the engine is running (even at idle) the oil is generally hot enough to lubricate properly. In this case the engine would have been warmed up, at first start the oil may be cold. According to continental the minimum oil temp for takeoff is 75F

Oil Temperature Limits

• Minimum for Take-Off .... 75°F.

On your average spring/summer day that may very well be ambient air temp. But we can take a look at their cold weather procedure (below 32 F ambient air temp on the ground) for the really cold situation,

8. When the oil temperature has reached 100°F. and oil pressure does not exceed 80 psi at 1700 RPM, the engine has been warmed sufficiently to accept full rated power.

So as long as your gauges are reading within those limits on approach I see no reason a go around would be any problem.

Please note the above listed info is for one make and model of continental engine and you should always consult your aircraft's POH for the numbers relative to your aircraft's configuration.

On a bit of a related note, when doing a touch and go you should apply throttle smoothly and not just slam it forward.

Do some flight schools really forbid touch & go's?

Not that I have ever heard of but some airports do (usually as a noise abatement thing) so it may just be that the flight school in question does because the airport does. That being said the planes generally belong to the flight school, they can make what ever rules they so please.

Are his concerns reasonable?

Eh, depends who you ask. In my humble opinion, no.

Is it really bad to make touch & go's for the engine?

Generally speaking, no. First off "shocking" an engine is generally referred to in the context of cooling it not heating it. For what its worth the consensus is that this is largely an academic debate anyway and may or may not actually contribute to engine wear. There is some logic that dictates that cold oil does not lubricate as well as hot oil but if the engine is running (even at idle) the oil is generally hot enough to lubricate properly. In this case the engine would have been warmed up, at first start the oil may be cold. According to continental the minimum oil temp for takeoff is 75F

Oil Temperature Limits

• Minimum for Take-Off .... 75°F.

On your average spring/summer day that may very well be ambient air temp. But we can take a look at their cold weather procedure (below 32 F ambient air temp on the ground) for the really cold situation,

8. When the oil temperature has reached 100°F. and oil pressure does not exceed 80 psi at 1700 RPM, the engine has been warmed sufficiently to accept full rated power.

So as long as your gauges are reading within those limits on approach I see no reason a go around would be any problem.

Please note the above listed info is for one make and model of continental engine and you should always consult your aircraft's POH for the numbers relative to your aircraft's configuration.

On a bit of a related note, when doing a touch and go you should apply throttle smoothly and not just slam it forward.

Do some flight schools really forbid touch & go's?

Not that I have ever heard of but some airports do (usually as a noise abatement thing) so it may just be that the flight school in question does because the airport does. That being said the planes generally belong to the flight school, they can make what ever rules they so please.

Are his concerns reasonable?

Eh, depends who you ask. Reasonable, maybe, correct, unlikely. It should be noted that thermal shock is a real thing and it applies to both the cold to hot and hot to cold situations. The jury is still out as to what effect this, if any has on airplane engines (in both cases). As noted in the comments to the question there are a lot of trainers out there who have performed countless touch-go's and have had no issues.