Do airlines measure baggage using linear inches or dimensions?

Which airline?
Many that make money on baggage have a sizer at the gate if yours happens to get flagged that you to past a test fitting it in before boarding. If it doesn't fit then be prepared to pay. Sometimes the aircraft is too small on certain routes for the bag to fit overhead. What makes matters worse you will see knucklehead passengers stuffing jackets and coats and personal items up there where bags should go .

Your bag is larger than the airline maximum dimensions. You should not use it.

Rule 1: The airline always wins

I think your argument is moot. 45 x 36 x 20 cm is smaller than any airlines dimension restrictions.

45 x 36 x 20cm is 17.7" x 14.2" x 7.9".

Which airline is 45 x 36 x 20 cm?

Many airlines use a "sizer box" that's basically a metal frame. If the bag will fit inside the box, you're good. It it doesn't fit, you can't carry it on.

Since any airline I've ever seen uses a sizer and not a tape measure I'd have to think it's the dimensions listed that count and not the linear distance.

Which airline is 45 x 36 x 20 cm?

EasyJet, unless you buy the Flexi fare which allows 56 x 45 x 20.

There are two categories of baggage that you can take on board :

Those that can be placed under the seat and those to put in the overhead bin.

For example with the budget airline EasyJet, the size limit: 45 x 36 x 20 cm is considered to be the maximum size for a cabin baggage to be placed under the seat.
With RyanAir this size limit is: 40 x 20 x 25 cm.

It's generally free

The cabin baggage to be placed in the overhead bin may have larger dimensions. (EasyJet: 56 x 45 x 25 cm, RyanAir 55 x 40 x 20 cm).

Airlines measure sizes before boarding. So it's important to read the baggage policy carefully when you buy a ticket, any excess will be charged much more once at the airport.

In general, US airlines are more lenient. But, when you board a flight at an EU airport, the gate agents are often not employees of the US airlines, but local airport agents. They may be stricter.

European airlines, esp. the LCCs, enforce the sizing restrictions much more seriously. They make lots of money from forcing passengers to check-in their bags at the airport.

I've been in a similar situation with my bag not quite fitting the airline's size limit. In my experience, if it's just a little over, they usually don't make a fuss about it. But, it can really depend on the airline and who you encounter at the check-in or gate.
I'd suggest giving the airline a quick call or reaching out to them on social media to see if they can give you a clear answer. Different airlines have different rules, so it's always good to double-check.

Flying in the USA or out of the USA the main airlines like Delta, American and United are not so picky, but when flying from Europe to another European city, they can be very strict.

We faced that a few years ago when flying from Heathrow to the USA with a stop at CDG Paris, they weighted our carry on bag, wife's purse and my briefcase. We exceeded the limit and had to pay to check one of our carryons.

Some airlines are also strict about the weight of carry on baggage. In Atlanta several years ago, Lufthansa had a woman who walked around the gate area eyeballing carry on bags. She stopped at quite a few and hefted them up to estimate the weight. A number of them got gate checked.

Simple answer --- Be prepared to meet the airline requirements. Lots of variation of enforcement. I have seen everything from strict weight at the checkin counter to size restrictions when boarding. If you are under their measurement, you are OK. If not, it is anyone's guess as to what happens. On a return home Delta flight from Prague saw a whole family - 4 kids - denied boarding because they had one carry bag that was too big and too heavy.

My Dad swears that if you use a soft sided bag that doesn’t look full then they will never ask you to size check it, and by using this technique he can carry more than the permitted amount of free luggage on EasyJet.

It’s true that he has never been stopped but I don’t know for sure that his bag wouldn’t fit in the sizer anyway.

In most US airlines, the regulation carry-on size is 22" x 14" x 9" or 45 linear inches. Those dimensions, HxWxT=2772 Ci (cubic inches). If you used just the 45 linear inches as the maximum, you could carry on a 15 inches (on a side) cubic box, which would have a volume of 15" cubed, or 3375 Ci, 20% more than 22"x14"x9". That would give you an extra ~600 CI of space in your suitcase. But at 15" high, it would probably not fit in the overhead bin, which is designed for a 14" bag on it's side. OK, then lets go with 14" x 15½" x 15½", which is still 45 linear inches but fits in a 14" high bin. Instead of 9" wide, at 15½ inches wide, these cubes would pretty quickly use up all bin space, and the airlines would but an end to that and hold you to the 22"x14"x9" sizer envelope.