I bought an S corp through a transfer of shares on 7/1/20 and the owner told me that only one of us should have filed taxes for the business in 2020 does that make sense?

Since ownership of the business transferred on 7/1/20, you should prepare a short-year tax return from that date until end of year and generate a schedule k-1 for yourself. The old owner should do a similar short-year tax return for the first part of the year from which a K-1 schedule would be generated for the old owner.

since you bought the S-corp stock the S-corp election remains in effect unless you revoked it by  filing a statement with the appropriate service center. In the statement, the corporation must notify the IRS that it is revoking its election to be an S corporation. The statement must be signed by each shareholder who consents
to the revocation (only shareholders at the time of revocation need sign so the former shareholder does not need to) and contain theinformation required by Regulations section 1.1362-6(a)(3).
if you continue the S-Corp only one return is filed for the year and in your situation both you and the former shareholder would receive k-1's based on the allocation of the various items on schedule K. . 

under reg 1.1377-1(a)(1) there are two methods

... each shareholder's pro rata share of any S corporation item described in section 1366(a) for any taxable year is the sum of the amounts determined with respect to the shareholder by assigning an equal portion of the item to each day of the S corporation's taxable year, and then dividing that portion pro rata among the shares outstanding on that day.

further (a)(2)(ii)

A shareholder who disposes of stock in an S corporation is treated as the shareholder for the day of the disposition. so old shareholder holding period includes 7/1/2020

my example business income for the year  $366 since there are 366 days in 2020 $1 is allocated to each day 

if 1 share outstanding the $1/day would be assigned to the 1 share. since the old shareholder held the shares thru 7/1/2020. you ach held the stock 183 days old 1/1 thru 7/1 and you 7/2 thru 12/31 so each would be allocated $183 of the business income.

the old shareholder either lied or did not know the tax laws. as long as it was an s-corp he gets a k-1.

now there is a second method set out in the same reg but that requires approval of all shareholders affected.

rules pertaining to the computation of each shareholder's pro rata share when an election is made under section 1377(a)(2) to treat the taxable year of an S corporation as if it consisted of two taxable years in the case of a termination of a shareholder's entire interest in the corporation.

In general. If a shareholder's entire interest in an S corporation is terminated during the S corporation's taxable year and the corporation and all affected shareholders agree, the S corporation may elect under section 1377(a)(2) and this paragraph (b) (terminating election) to apply paragraph (a) of this section to the affected shareholders as if the corporation's taxable year consisted of two separate taxable years, the first of which ends at the close of the day on which the shareholder's entire interest in the S corporation is terminated. what this means is that the books are closed on 7/1/2020 if the business income was only $100 thru that date it would be allocated to old shareholder. the other $266 ( remaining income from above example ) would be allocated to you. If the event resulting in the termination of the shareholder's entire interest also constitutes a qualifying disposition as described in § 1.1368-1(g)(2)(i), (it does) the election under § 1.1368-1(g)(2) cannot be made. An S corporation may not make a terminating election if the cessation of a shareholder's interest occurs in a transaction that results in a termination under section 1362(d)(2) of the corporation's election to be an S corporation. (See section 1362(e)(3) for an election to have items assigned to each short taxable year under normal tax accounting rules in the case of a termination of a corporation's election to be an S corporation.) A terminating election is irrevocable and is effective only for the terminating event for which it is made.

(2) Affected shareholders. For purposes of the terminating election under section 1377(a)(2) and paragraph (b) of this section, the term affected shareholders means the shareholder whose interest is terminated and all shareholders to whom such shareholder has transferred shares during the taxable year. If such shareholder has transferred shares to the corporation, the term affected shareholders includes all persons who are shareholders during the taxable year.

(3) Effect of the terminating election -

(i) In general. An S corporation that makes a terminating election for a taxable year must treat the taxable year as separate taxable years for all affected shareholders for purposes of allocating items of income (including tax-exempt income), loss, deduction, and credit; making adjustments to the accumulated adjustments account, earnings and profits, and basis; and determining the tax effect of a distribution. An S corporation that makes a terminating election must assign items of income (including tax-exempt income), loss, deduction, and credit to each deemed separate taxable year using its normal method of accounting as determined under section 446(a).

while the first method requires no special statement this termination election requires the following

To make the election, the corporation must attach a statement to a timely filed original or amended Form 1120-S for the tax year for which the election is made. In the statement, the corporation must state
that it is electing under section 1377(a)(2) and Regulations section 1.1377-1(b) to treat the tax year as if it consisted of 2 separate tax years. The statement must also explain how the shareholder's entire
interest was terminated (for example, sale or gift), and state that the corporation and each affected shareholder consent to the corporation making the election. A single statement may be filed for all terminating elections made for the tax year. If the election is made, enter “Section 1377(a)(2) Election Made” at the top of each affected shareholder's Schedule K-1.