Although Berlin has insisted the rallies were cancelled by local authorities for logistical reasons, Turkish officials have hit back repeatedly, with Erdogan even comparing such actions to "Nazi practices".
In an attempt to maintain its relations with Ankara -- a crucial partner to halt the influx of refugees and migrants into Europe -- Germany has refrained from further exacerbating the tensions, urging Turkey to "keep a cool head".
But the spat has provided fodder for those working for a 'Yes' vote in the April referendum.
"This tension serves Erdogan's referendum campaign," said Ahmet Insel, a political scientist and academic.
"They have difficulty finding themes to fill their campaign and have difficulty mobilising the 'Yes' camp," he said.
According to Insel, the move to prevent such rallies in Germany and elsewhere in Europe has allowed Erdogan to resort to "the discourse of victims and (to) fuel anti-European resentment".
The academic said that 300,000 to 400,000 'Yes' votes from Turks abroad could represent a difference of one percentage point in the referendum, "especially if participation was lower in Turkey".
In the 2015 elections, the ruling Justice and Development Party (AKP) which Erdogan co-founded, won the backing of 60 percent of Turks living in Germany.
Back home, it secured just under 50 percent of the vote.
Erdogan's AKP is running the 'Yes' campaign, saying the changes would bring political stability and avoid the potential confusion from a two-headed system with both a president and prime minister.
While the outcome of the referendum seems uncertain, the government is trying to "seize every opportunity to gain a political advantage," said Sinan Ulgen, head of the Istanbul-based Centre for Economics and Foreign Policy (EDAM).
Turkish leaders attach an importance to these rallies, Ulgen said, not only because of the influence on Turkish voters in Germany but also because, through the media coverage, they "can possibly influence the nationalist vote in Turkey".
Germany is home to the largest Turkish diaspora outside of Turkey with 1.4 million people who are eligible to vote in the referendum.
Although the community only represents about 2.6 percent of Turkey's 54 million eligible voters, it represents an attractive target for campaigners.
But Erdogan's latest diatribes recalling Germany's Nazi past may have been a broadside too far, Ulgen said.
"It is not certain that the (Turkish) population, and especially the economic circles -- since Germany is a major trading partner for Turkey -- is inclined to support rhetoric as fierce as the comparison with the Nazi regime."
'Isolating itself... again'
Jean Marcou, a Turkey expert at Sciences Po in the French city of Grenoble, said the referendum's outcome could be tighter than expected.
He said Erdogan's strategy of "defiance" could cause concern among some parts of the electorate "because Turkey is, again, isolating itself completely".
"Erdogan enjoys the dispute with Germany. He seeks external enemies useful in domestic politics to win his referendum," Omid Nouripour, an MP and foreign policy spokesman for the German Green party, told Germany's Bild daily.
Turkey has also been sharply criticised by other European countries including Austria and the Netherlands for seeking to campaign abroad.
Despite the furore, two Turkish ministers have visited Germany in recent days in defiance of European opposition and the prevention of meetings in which they were to have taken part.
"The government could ban the arrival of Turkish politicians, but would that be smart to give Turkey the role of martyr and to give Erdogan the voices missing for this referendum?" asked Julia Kloeckner, vice president of the Christian Democrats (CDU), Chancellor Angela Merkel's party.