I came across this article while researching about the Christmas truce of 1914. To save you some time looking for the paragraph, it talks about the Princess Mary Christmas Box beside the picture of the two soldiers, and then the claimed German counterpart in the next paragraph.
Not to be outdone, Fritz received a present from the Kaiser, the Kaiserliche, a large meerschaum pipe for the troops and a box of cigars for NCOs and officers.
Is this true? This is the only article (that I've found) that states the German soldiers received a gift just like the British did.
The answer appears to be yes.
With regard to the specific German counterpart to the Princess Mary tins, Stanley Weintraub (in Silent Night, 2002; p. 12) has this to say:
The kaiserliche equivalent to the Princess Mary box was, for the ranks, a large meershcaum pipe with the profile of Crown Prince Friedrich Wilhelm on the bowl, or a box of cigars inscribed Weihnacht im Feld, 1914. Noncommissioned officers got a wooden cigar case, each inscribed Flammenschwert -- a flaming sword.
Beyond that, though, the bulk of charitable presents received by German infantry on the front were typically called Liebesgaben, or "loving gifts." These were collected and distributed regularly in the early stages of the war, and usually took the form of candies, sugar, cigarettes and tobacco, socks, and so on. There was an especially vigorous push for them for Christmas of 1914.
Jeffrey Verhey, in his The Spirit of 1914: Militarism, Myth and Mobilisation in Germany (2000), provides (p. 106) some numbers for this distribution of Liebesgaben, though I'm not sure if the figures in the latter half of what I cite below are just for one transport hub or for the whole of the program:
The city government of Frankfurt/Main required fifty train wagons in order to transport its gifts.
[...]
In HStA Nordrhein-Westfalen, Regierung Dusseldorf, Prasidialburo no. 1208, there is a list of Liebesgaben sent to the front. It has 273 pages. In December 1914 alone 22,090 shawls, 26,188 shirts, and 32,743 pairs of socks were sent to the soldiers.
Returning finally to Weintraub, he quotes a number of German soldiers describing their Christmas hauls -- the gifts distributed by popular charity rather than on the Kaiser's behalf appear to have been far more varied, bountiful and well-received, but some soldiers (much like their French and British counterparts) found the whole exercise more of a nuisance than a pleasure: the distribution of these Christmas parcels along the line created enormous logistical problems for each participating army, in many cases disrupting the delivery of day-to-day war supplies.
If the official German infantry gifts came in their own special tin or box, I don't have any information on that. I hope the above is of some help all the same.