Let's get the easy question out of the way. Is this depiction created by/for the anime? No it is not. The exact same depiction can be found in NHK's 1987 Taiga Drama Dokuganryū Masamune, and maybe the novel it is based on. The source of the story is likely based on the Ujisato-ki.

However did it actually take place? This is what the Ujisato-ki says:

...其時政宗風情コソ聞モ恐シケレ死装束出立テ金箔ヲ押タルハタモノ杭ヲ馬ノ先持セテ上洛アリシト聞ヘシ其跡蒲生朝臣モ打立上洛有...
We heard Masamune at the time was so scared he set off wearing burial clothes and had a gold-gilded crucifixion stake carried in front of his horse and set off for the capital. Afterwards Gamo Ason [Ujisato] also left for the capital.

I would personally interpret that as Masamune riding his horse and had other people carry the cross in front of him. Masamune set out from Yonezawa and took 24 days to reach Hamamatsu, a distance of about 550km. He then met Hideyoshi four days later at Kiyosu, a further 115km or so, before proceeding on to Kyōto (Hideyoshi had been out hunting and arrived back in Kyōto a day before Masamune). Kajūji Harutoyo recorded in his diary Masamune had with him over a thousand men when he entered Kyōto, so this was around the correct speed for a normal foot march. This means there was no way one person carried a crucifix (I'd expect that to be damned heavy) the entire way. Now that of course doesn't rule out a group of people carrying the crucifix, but...

The more important thing is the word heard. The author of the Ujisato-ki admitted he didn't actually see it, it was just something he heard. We don't actually know who wrote the Ujisato-ki, only that it's a record centering around Gamo Ujisato. The Ujisato-ki itself says it was written between 1633 and 1637, meaning it was written at least 40 years after the event. And that is assuming the author didn't make up those dates. Even if said author was really alive at the time and actually heard such a story, by his very own words he would have been at Aizu or Nihonmatsu, and no one from the Gamo clan actually saw Masamune and his burial clothes and golden crucifix as the author records himself that Gamo Ujisato only set out for Kyōto after Masamune had already left. By the way Ujisato actually left Aizu three or four days before Masamune left Yonezawa. And as Yonezawa is over three days march north of Aizu, Ujisato would have arrived in front of Hideyoshi a week before Masamune, so Ujisato-ki already got that detail wrong.

The crucifix and burial clothes are not found in Harutoyo's diaries, or that of Nishinotōin Tokiyoshi, or Yoshida Kanemi. You would think such attention-grabbing display would, you know, grab attention. Masamune's own letters from around the time make no mention of them. Twenty days after Hideyoshi passed his judgement, one of Masamune's retainers wrote back to the other Date vassals at Yonezawa about what's happening in Kyōto. He plays up the Date name, saying lords and commoners alike in Kyōto are awed at the clan's martial deeds, and yet does not mention the burial clothes or crucifix let alone how they were received. It is also not found in other Date or Gamo records I can find (admittedly the Gamo clan left little records). According to the Maeda clan (recorded in the early Edo), when Masamune and Ujisato were arguing in front of Hideyoshi (Maeda Toshiie was present) Masamune was wearing a Kamishimo, so not a burial garb. I can find no other mention of the story in historical sources, not even in Edo-era sources. Even high-late Edo historical fiction make no mention of burial clothes and included the story of the golden crucifix basically as an endnote, further explaining that "whether the story is true awaits the debate of those knowledgable." And this is a fiction which depicted things like Masamune attempting to poison Ujisato, the two of their forces almost coming to blow (both probably also sourced from the Ujisato-ki), Katakura Kagetsuna telling Masamune to calm down and travel to Kyōto to see Hideyoshi ASAP, and that they brought with them one and a half typed pages of gifts to smooth relations, and included the story of Masamune getting out of trouble by insisting the letter being use as proof of his guilt was false because the signature doesn't match. In other words, the burial garb and golden crucifix was less believable than all of the above.

So what do we make of the story? Well we know tensions between Ujisato and Masamune ran so high that Ujisato publicly accused Masamune, with proof, of inciting the Kasai Rebellion. If we focus on that even that Ujisato-ki treats it as a rumor, it's most likely something the Gamo men made up, either at the time or afterwards, to laugh at the state Masamune found himself in, as payback for all the problems he caused them (and himself).