Have national symbols often acquired partisan meanings?

I would argue that at least in the United States southern conservatives traditionally drape themselves in the flag to show their patriotism. There is a public display factor in the practice. Perhaps this is the nature of conservatism, which cherish history and it's symbols rather than more forward looking expressions of patriotism. Perhaps this is a genuine attempt to appeal to all Americans across the political spectrum calling upon a symbol both parties share and honor. Perhaps this is a legacy of Southern Conservatives who still have a bit of a chip on their shoulder over their once lapse in patriotism from the mid 1800s. An attempt to inoculate themselves from patriotic criticism on historic grounds. Likely it's more than a single reason depending upon the individual. Still when Southern Conservatives were Democrats the Democrats were the party of the flag.

• Democratic President Woodrow Wilson in a proclamation first established the flag holiday in 1916.
• The Flag Holiday was formalized by an act of a Democratic Congress in 1949. (81st Congress Democrats controlled both House and Senate )

Southern Conservatives left the Democratic Party due to deep disagreement with President Johnson's policies in the 1960's, and the GOP became their new home, and the new party of the flag.

In 2008 Presidential run, then Democratic Senator Barak Obama faced repeated questions over his patriotism including a question during a national televised debate (April, 2008) hosted in Philadelphia over his lack of an American Flag lapel pin.

Pretty much every national symbol stems from partisan/patriotic sentiment. That's how they became national symbols in the first place, they have over time become synonymous with a country.

So in your specific case (as in most all of them) it's the other way around from what you're trying to say, the US flag isn't partisan, it's a symbol of patriotism and the left has lost that, they're no longer patriotic and therefore have abandoned the flag rather than the right adopting it. It would be silly for a democrat to walk around with a US flag while screaming how bad the USA is and how disgusted she is to be an American (I've seen it way too much), just as it would be silly for a republican to wave around a communist Chinese flag and shout how much they love communism.

Let me present a specific example from Central Europe.

The Árpád-stripes:

Originally it was the heraldic device of the Árpád-dynasty. Of course there were no nations of the modern sense in the medieval times, and the Kingdom of Hungary had a different flag, but still, considering that the Árpáds ruled Hungary for more than 300 years, it became quite a symbol.

It even made into the coat of arms of modern Hungary:

Yet, after it was part of the banner of the Nazi-like fascist Arrow Cross Party during the Second World War, it was never officialy used in its "pure" form.

It is nowadays used by right wing nationalists, who think it embodies an old tradition of Hungary, and claim that the form used by the fascists was different. (it had 9 stripes instead of 8) But you would never see a leftish Hungarian wearing it it.

National symbols are always non-partisan; but from time to time one party or another may wish to disassociate itself from them. In 2012 the Democrats were not the least shy about use of the American flag, because they were in power.

Since the 2016 election, Democrats have deliberately shied away from using the flag. This is a means of expressing displeasure with the current administration, and is done presumably because it plays to their base. It is emphatically not that the Republicans have come to own the flag and other national symbols, but that the Democrats have deliberately elected not to use it at this time.

It's not that national imagery has acquired a Republican flavour - but that Democrats have (temporarily) lost a taste for national imagery.

As a refutation of your claim "but it is sort of owned by one side." i submt the aftermath of the 1965 Canadian flag debate Then PC Leader John Diefenbaker made much of the Liberal choice not to incorporate "PC blue" in the flag, choosing instead a flag exclusively white and "Liberal red". Once the flag was officially adopted however, the other major parties quite contentedly accepted the imagery:

Then NDP Leader Jack Layton addressing Party Convention:

Leaders Peter McKay (PC) and Stephen Harper (Reform) announcing the merger of their parties:

Note: Yes, it's not a coincidence; in Canada you can almost always identify a politician's party by his tie; particularly at important events or occasions. Note Layton's orange tie, McKay's blue tie, and Harper's green tie in the pictures above.